I have lived with birds in homes heated by antique radiators (water) and natural gas heaters. If you use natural gas, a carbon monoxide detector is important (as is frequent upkeep).
There are some hot-oil radiators that are recommended for birds as alternatives to more dangerous space-heaters and wood-burning stoves. The hot oil heaters can be found on bird websites---It's all internal (no fumes) but do make sure you research first...I am not sure they are all created equal (if that makes sense..)
Basically, you don't want anything that produces any sort of exhaust/fumes--- even candles and oil warmers are dangerous to birds' sensitive respiratory systems.
https://birdtricksstore.com/blogs/birdtricks-blog/bird-safe-heaters