I sympathise with you totally as I have been trying (without success) to order online from Sainsburys for 2 weeks for my aunt in Wiltshire. Firstly I wonder if you have a separate account for your parents? or have added them to yours? The reason I ask is that I suspected proving I was shopping for my aunt might be a hard thing to do on my own account and so set up a separate account for her which I later realised I needed to link to her Nectar card as I thought her age/DOB on her Nectar registration would be needed to enable orders with deliveries. But it seems it's more complicated than that! In fact a minefield as you no doubt are discovering.
A bit of context.. My aunt is 80+, very immobile/ disabled, has had cancer though not having treatment now, and has a raft of other underlying health conditions including borderline diabetes. You would think this would mean she is classed as vulnerable but she did not get the letter confirming she is recognised as a vulnerable person eligible for home deliveries!
What I saw on her account was what you describe no delivery slots at all initially and then at the end of last week I got the option to do a click and collect only from a supermarket 40 miles away! As my aunt can't drive any more and the distance would be too far for any kind neighbour/volunteer to travel this is a complete nonsense. Sainsburys says on their site that if anyone believes they (or someone they are shopping for) are a vulnerable adult not recognised by their system then to call an 0800 number. All last week from morning until night this number was engaged. I sent a message and got replies but very frustratingly all the messages told me how busy things are and please keep trying the 0800 number. At the point of giving up I received an email giving a link to a government website on my account where I could check my aunt's vulnerability status. I heaved a sigh of relief that I could do something and in my mind once I entered her details I was anticipating (hoping) that the settings on her online shopping account would be changed to enable deliveries. Not the case at all! What shocked me was that the classifications for 'vulnerable' (and therefore eligiblility for deliveries) are very narrow - medical conditions that make you vulnerable to getting corona virus - that's it - or receiving certain cancer treatments and/or having particular types of cancer - all very valid reasons. But vulnerable does not mean as I had thought those who are also elderly, live alone and are registered disabled with other medical conditions who urgently need home deliveries too at this critical time. As people all around self isolate or choose not to travel so far for shopping and are having difficulties getting their own provisions never mind someone else's this is very anxious making. And for relatives in London at a distance trying to help even more. So I am still not any closer to resolving this issue. The really worrying thing for me is that it seems that the only thing that my aunt can get delivered apart from milk is alcohol! by the box load which is making the whole situation worse and I won't even go there as to what that means! I'm sorry not to offer any solutions but maybe someone somewhere with an influence over these matters may see these posts. It is a very challenging and worrying time for us all and especially our vulnerable family members. If you find a way to resolve the situation for your parents I would love to hear more..