You won't be able to convince them to move Heaven and Earth just so you can play one game on a PS4.
They're going to say save up enough money to get a PS4 and a new TV. That's the logical answer. Or you should be able to find an adapter that converts the signal to RCA jacks. That's what we did with the PS3.
Ask random questions, just sit with them and watch tv and as questions about EVERYTHING until they get tired of you, then tell them why you are there instead of playing games.
Why do cow babies have to be coached out?
Why is there hair growing in areas im not used to hair growing?
Why do parents mention bees during the sex chat when bees do not have sex?
Are we there yet?
Tune in next time for another glorious episode of "How not to get what you want"
isn't this exactly the situation for which remote play for the vita is designed for?
just buy a 200$ vita, hook the PS4 up to your parents TV, turn it on, put a game in, and play it on that beautiful vita screen, sure it's a little small, and you can't use the dualshock 4, but it's better than nothing I suppose
Tell your parents this........