can parent private key sign a transaction created by a child address? like this: 

I have a private key P, and via a HDchain, it creates a child address addrChild with a child key pair: priKeyChild and pubKeyChild.

When I create a rawTransaction with the addrChild, I can successfully sign this rawTransaction with the priKeyChild. 

the question is: instead of using the priKeyChild, can I successfully use P to sign this rawTranscation ?
No, you cannot. The relationship between parent and child keys is via HMAC-SHA512, not related to ECDSA (the signature algorithm). To ECDSA, the parent and child keys have no relationship, they are both just private keys. So only the child private key can sign for the child public key, not the parent.