It partly depends how much you want to spend, and also how long you expect to be at the same flat.
I can't say I'm a fan of having the phone line and broadband bundled together. It normally means you are tied in more than you are by having them separate.
Ignoring that, on that exchange you can get Be. That is technically the same as, slightly more flexible in fact than, the old O2 packages your parents are on. That would be my choice, with the line from BT on an 18-month contract costing only £30 installation. You need to make 10 calls per month I think, but those can be within the included freebies. So you could make dummy calls to your parents, (or anywhere else), one after another, once a month.
Others to look at for the line are the Post Office and Be. The trouble with Be is that if you migrate out you have to change line provider as well, and that gets messy. Hence what I said at the start.
You could get both from Plusnet, but you would need the "Pro" "Boost". That prioritises gaming and also solves the traffic management, which is set out on
this page.
What didn't you like about Sky? For the broadband side it is next best to Be.
My broadband basic info/help site - www.robertos.me.uk
My domains,website and mail hosting - Tsohost.
Internet connection - IDNet Home Starter Fibre.
Live BQM.
"Where talent is a dwarf, self-esteem is a giant." - Jean-Antoine Petit-Senn.