Anecdotally, parrots don't know to fly off if their perch gets too hot.
Cue "charmedbyekkie is a bad parront" story:
Cairo likes to sit on our tall kitchen faucet, so he can bend down and drink straight from the tap. I normally leave it on cold for him, but to be honest, leaving the temperature dial in the middle usually also gets you cold water. So I left it in the middle.
And out of the corner of my eye, I saw Cairo start to shake one leg vigorously, then the other leg. He started alternating shaking his legs.
I rushed over and swept him off the faucet, apologising profusely. The faucet, for some reason, decided to output hot water, and the faucet itself was hot to touch. I then cranked the faucet all the way to icy cold and ask him to step back on to try to help his poor feet. Ultimately he was fine and back to normal within minutes (I kept an eye on his feet, and luckily the water hadn't been boiling hot).
So yep, I'd recommend not letting him sit on the heater. He won't know what to do when it gets too hot. Like a frog in water starting to boil.