Currently I am doing these:

- Hanging knee raise on rings (top position hold) - 5 sec. hold per rep, eventual goal is 3 sets of 12 reps. This is preparatory towards an L-sit, which I can't do yet. Once the hanging knee raise goal is achieved, I'll start on the next difficulty level, which is single-leg hanging straight leg raise. After that is the double-leg.
- Skin the Cat on rings.
- Tuck Sit to Tuck Bent Arm Stand on parallettes.

It's true all the ring exercises hit the core one way or the other, but these are the ones that hit them hardest for me.

In the recent past, I was doing tuck swings, assisted inverted pushups, plank-to-tuck swings, one-leg shoulder stand holds, mountain climber holds, etc. on the parallettes. Those all hit the core too.

I learned all these exercises by following GMB's Rings One and Parallettes One programs. Each has fairly clear progressions from really basic exercises (eg. pushups on knees, reverse rows, assisted pullups (feet on ground), etc.) to ever more difficult ones.