Learning k-parities with Membership Queries and Persistent Noise
If I understand correctly, this is exactly equivalent to local decoding of the Hadamard code.

We can think of the oracle (with the noise) as being chosen beforehand, and after that we can view the oracle as a corrupted codeword of the Hadamard code. Learning the parity in this case is exactly the task of local decoding the original message from the corrupted codeword.

This is well-known to be possible using $O(n \log n/\delta)$ queries, using the Goldreich-Levin theorem, e.g.:
https://lucatrevisan.wordpress.com/2009/03/09/cs276-lecture-12-goldreich-levin/

On the other hand, it clearly can not be done using $o(n)$ queries, since the algorithm has learn $n$ bits about the parity - i.e., its attributes.