I view my DH and myself as a team. 
I help with his surviving elderly parent, sometimes more than his sister but it's because of how I view my DH and our marriage.
Parent once had a discussion with my DH pondering whether I loved parent more than parent's own daughter does. 

I note an earlier comment asking in the OP will do same for his wife. 
In my view fairness, equity or equality isn't sameness and seeking for sameness isn't usually the best for the team and even the individuals that make it up.
My DH does and gives of himself in certain ways that I won't take easily to or isn't best for our team and I give of myself or do other things in ways that is best for our team. 

I view my DH and myself as a team. 
I help with his surviving elderly parent, sometimes more than his sister but it's because of how I view my DH and our marriage.
Parent once had a discussion with my DH pondering whether I loved parent more than parent's own daughter does. 

I note an earlier comment asking in the OP will do same for his wife. 
In my view fairness, equity or equality isn't sameness and seeking for sameness isn't usually the best for the team and even the individuals that make it up.
My DH does and gives of himself in certain ways that I won't take easily to or isn't best for our team and I give of myself or do other things in ways that is best for our team. 

I actually disagree with the majority of posts on here. I think life pulls us in all sorts of directions, and as a couple you can work together to jointly support the areas of life that need it (and I would include parents and parents in law as being part of that life). It sounds like you are struggling and need help supporting your family so think you should have an honest discussion with your wife and see where you as a couple can jointly flex your other obligations to be able to support your parents jointly. For example - skip the play date, sports event or dinner party for a bit. By not doing this there could be unintended consequences that impact both of you in other ways (e.g. you get unduly stressed; you don’t perform at work and so miss your targets/bonus…etc). My husband and I work extremely long hours, and full time, and there are always compromises we both have to make to make it work and also support our wider family (sometimes he is more busy than me and sometimes vice versa). In addition; I can imagine it is really upsetting seeing your parents health deteriorate as well as causing the time challenges you mention, so definitely worth the conversation and support to ensure she understands your needs and you work out a solution together.

Thanks everyone.

I think the main takeaways are that although all families are different it's not as common for spouses to help out on a regular basis with in-laws care, emergencies outside. I also like the comment about not trying to change her view.

Thank you everyone.

We are all different.

For me - it’s v clear - each looks after their own parents, that’s a responsibility shared with your siblings, not your husband.
I don’t think I could entertain the “doing it as a team” - not because i think it’s in any way wrong, but it just doesn’t resonate with me. That’s just me, it’s of course a personal choice, depends on how close you are with in-laws, your upbringing, etc etc.
Where I would be vey happy to help is - do more for the children/our household if my husband has to spend more time looking after his elderly parents. But his work commitments vs care of his parents - are definitely his to manage, not mine.

You should probably discuss with your wife so you each understand where you stand on this. But if she pitches in looking after your father, you would have to reciprocate and help look after her parents.
I would also not look to change your wife’s outlook on this, as it can only lead to disagreements between you.

Aging is a difficult concept to deal with and it can be stressful for the parent and the child.

I suspect I am much closer to your father’s age than yours. I would not want my children or their spouses too involved with my day to day care.

You suggest your wife’s reluctance to help may be ‘learned behaviour’ with regard to her own family relationships. I would suggest you may well need to look at your own ‘learned behaviour’ which seems to suggest it is a given that your wife is obliged to take care of your father.

Thank you for all your replies. There is a lot to take in and I'm also working hard to make sure that I don't assume you should help just because it makes my life easier. As I said I think part of the issue is that she's just not that close to her own parents and so some of it may be "learned" behaviour.

I"ll read the replies in details and then revert.

Thanks again.

Both myself and my partner have recently experienced this before our parents died. We both took responsibility on our own alongside our siblings and do not help each other. It is a big ask and it’s probably better just to take on the responsibility yourself. However have a proper talk about it so it doesn’t remain an unspoken resentment but my advice would be not to expect anything from your wife here. Now that my mum is gone I really cherish the memories of caring for her before she died including time spent at appointments etc. my mum also enjoyed sharing these moments. I know it is very hard to manage all of this and it can feel overwhelming so I do feel for you.

What other tasks does your wife do? Does she also work? Look after children? Manage the household? I ask because a lot of the work women do is unseen &  under valued, so it may well be that your wife is simply too busy to add "Look after husband's elderly parents" to her already overflowing list. 
Be honest - do you always approach all tasks "as a team", or are you asking her to make sacrifices to help you when you don't do the same in return? If you have demonstrably helped her with "her" tasks in the past, then, yes, I would expect her to pitch in. However, if she feels like the same isn't true when roles are reversed then that could lead to some reluctance to assist. 
I realise I may be projecting, though, as my ex husband would watch me drown & not lift a finger whereas I was expected to ask "how high" as soon as he wanted me to jump, but to me it sounds as though she's digging her heels in for a reason. Also, I wouldn't compare her to your SIL, as I can't imagine that having a positive effect on her desire to help. 
 

I agree with all the posts above as this is a situation where there is no rules to follow it is up to different individuals and differenr relationships.

In general i dont want to be involved in the day yo day caring of my parents in law even though I have a great relationship with everyone. When I m there ai cook and I shop and help with tidying etc and generally dont my MIL do anything.
However, in this situation i would absolutely help.
I would see it as stepping up to support the family as i m married to my husband and therefore i belong to a family as well as mine, and in our families we support and help each other.

So while i dont want them to expect my help, i of course give my help when needed.
I love them all (even when they can be difficult).

I would equally help a friend, etc. It s the values I want my children to grow with.

Agree with most of the above comments.

Every family is different, with different personalities, skills, abilities and so on. Some are averse to hospitals or afraid.
Not every family works 'as a team' , ive heard this phrase a lot and doesn't resonate with everyone.

Be philosophical and accept this, make an individual request on an ad hoc basis , don't expect a regular 'show up' like an obligation.
I do agree its your Dad not hers. She may develop some sympathies, like a first-degree relative, if she got to know him a little more, if she did it occasionally .
Perhaps go together with your wife, the first time?

Do you help her with her family commitments? It may be she's just following your pattern?  If you do help her with her family or other obligations, I think you should have a chat about this and ask her why she's not reciprocating. 

Just because your sister in law behaves in a certain way does not mean your wife should. They may have had a closer relationship over the years and you sister in law lives closer/has more time/is happy to do it.

As you say your father resides in a care home the simplest solution would be for them to arrange for someone to accompany him. As they see him day to day I can’t see why he would be uncomfortable with that arrangement. The extra charge would be worth avoiding family resentment.

Both my parents died many years ago and I would help and have offered many times to take my partners parents to appointments etc.
I have a good relationship with them.
His siblings help too.
I view us as a team and if one person has time then stick your hand up!
Obviously do no where near as much and don’t feel an obligation but would want to help my partner thus if I can help his stress then why wouldn’t I do that

Hi
Rule of thumb here is each of us deals with their own family and takes care of parents with sibling(s).

There are things we, as children, can say to our parents, that a DIL or SIL cannot say to their in laws or wouldn't be comfortable dealing with. As they age parents could be difficult and we would never want to put each other in a difficult position with in laws.

I would never ask my husband to help and equally I would never offer to help with his either. We both believe it would only create tensions. However I'm here to support my husband in any ways when he needs to do that (and the opposite is true as well) : for example by being fully in charge of kids, home, school - any aspect of our life's family - when one of us has to visit and take care of them, whether it means a day or a week.