What ‘support’ are you expecting?
They are supporting you by allowing you to still live in their home as an adult.
You want to spend your extra day off hanging around at home, with them.
You say you don’t go out so you’re always there when not at work.
This sounds like the first time they’ve expressed any negativity that makes you feel unwelcome.
My DD and her husband lived with us for a long time. I love them both dearly but it is a different dynamic with 4 adults living under the same roof. Would I let them move back if their circumstances needed it? Of course I would but I’d much prefer not having to.
DH and I went through a phase of trying to watch an old classic film every Friday night. We’d get our dinner and settle down to watch it. DD would come home from work and want to discuss all her work worries the second she got through the door. So we’d pause the film and listen, chat it through with her, then when she’d run out of steam we’d press play, then she’d start again, so we’d pause it. Eventually she’d get up and go and make her dinner. So there would be cooking noises and smells in the background. Her now DH works shifts. He’d come home very late and as much as he’d try to be quiet and respectful I’m a light sleeper especially if I know someone is not home. He’d also sometimes cook something when he got in so they’d be cooking smells at midnight.
None of this would make me not offer them a place in my home if needed. But it’s much calmer now it’s just the 2 of us and when they visit or FaceTime we are very happy to have them and give them our full attention.
Its not that your parents don’t want you, it’s that they are starting to enjoy their own space and the thought of losing another day of that was concerning for them. you however are acting like a stroppy teenager who would cut them off for continuing offering you a space in their home but panicking a bit when they realised they’re losing another day of peace.