I was parking facing up a steep hill and didn’t engage/pull up the parking brake enough to keep the car from rolling backwards.
Would this do any damage to my transmission?
It is usually best to park using the parking brake — but not relying on the parking brake alone to prevent a roll-away.
The transmission should always be put into park for an automatic and reverse for a manual.
The third needed preventative measure is to park near enough to the curb that the front wheel can be turned outward about 40 degrees when parking up-hill and the car allowed to gently roll back into the curb until the back of the front tire gently bumps into the curb. Then place the car in gear and shut down the engine.
When parking down-hill, turn the wheel toward the curb about 40 degrees and allow the tire to gently roll into the curb.
Automatic transmissions generally are made strong enough to hold the car’s mass in parking gear, but there is no reason to test your transmission in this way.