Parking pins don't connect to the flywheel, and an automatic does not have a
flywheel, just a flex plate and torque converter (a flywheel equivalent).
More likely is that the transmission oil pump has finally worn out, or the
torque converter has failed.

Check your transmission oil level just in case. Check that the lever on the
top of the trans is actually moving into the correct place.

An auto transmission shop can evaluate the transmission. It could be a
simple problem, but I'd say age, as your car is 23 years old and probably
done at least 300,000 km. That's pretty good for an automatic. Rebuilding an
old  trans PROPERLY is quite expensive, and getting all the parts for an '85
may be problematic.

If the body is very sound, then its worth spending $2000 to fix it. Just
make sure that you get a decent warranty (at least 6 months) and understand
what is going to be done. Just flushing it and dropping in a pump and valve
body kit is NOT going to be enough. There's a variety of bearings, seals,
possibly a valve body hat are due for replacement.

Your engine is also very tired at 300,000km+. The head in particular will be
in need of a reconditioning.

Get some quotes. You may find its WAY cheaper to just buy a newer vehicle.
Here in Canada a decent 2002 Lancer can be had for about $4500.