The parcels are a joke so no advantage in that dept., I can assure you. I insisted I didn't need one yet ended up with 3 which I managed to rehome with other people. I'm coeliac so there was nothing in the boxes I received that I could eat.
I too was wrongly added to the government's food parcel list and received two boxes, a local councillor was able to make sure they went to someone in need. But having been a vegan for over twenty years there was little for me in them either. Something I told the government helpline when they asked if I wanted them, I said I only needed supermarket access, but they said local authorities can arrange ones for people with dietary needs.
Of the supermarkets only Morrisons have not given me priority access. Looking right now the earliest slot I can get from Tesco is Wednesday 6am – 2pm, from Sainsbury's is 8:30am – 9:30am tomorrow, Waitrose is next Monday at 6am, and Asda is 7am on Thursday week.
The delay with Waitrose might be because I used them on Saturday, and they only want you to use them weekly. As if I could afford that often! Asda are the only difficult one for me to get a slot with, which I think is because they waive the minimum order and delivery fees for priority access so must get overloaded.