I would immediately turn it off & unplug it. Then, remove him and put the heater away where he can't perch on it. To avoid a habit developing where it could eventually become a hazard. Better to be safe than sorry.
We have two wood burning stoves. From the beginning I taught him and reinforce that they are Hot & Dangerous. He NEVER lands on them not even in the warmer months. Thank God!!
Don't start that habit---maybe consider getting one of those heated perches that they sell for birds---he likes the heat, but he could easily ignite....I had a heater ignite in my lap when I was in high school (we had an unfinished basement and I had it on me for warmth--much like your bird---anyway, it just mostly scared me, but it would have been bad for a bird and my incident could have caused an electrical fire looking back on it....) It was like my body prevented enough heat from leaving the unit, so the unit got too hot (just a guess) and literally started burning. A tail feather or something like that would burn easily without the bird noticing right away that it was too hot, so I would definitely move this thing and find an alternative. Imagine the damage if he got his toe stuck, or it caught fire.
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.
Radiators still can get very hot during certain times of the year, so I would say yes---also, because things sitting on radiators can heat up and start of off-gas (like plastics etc).