You can prove that you know c_par such that both (I_L,I_R)=hash(c_par, K_par, i) and K_i=I_L*K_par, without revealing c_par itself, via zero-knowledge computational integrity (PCP) proofs.
Really, you can zero-knowledge prove that you know a preimage of a hash?
Yes,
you don't send the entire proof (this wouldn't be ZK and the proof is long anyway), you just let the verifier query a few bits from the proof, and if you didn't know the preimage then the verifier will catch you w.h.p.
Edit: actually you can send a very short full proof by building a Merkle tree from blocks of the long PCP proof and using the Merkle root as seed for pseudorandom queries, see SNARKs for more info.