Is AES's parity key-dependent? Is the parity of the permutation of the set $\{0,1\}^{128}$, defined by AES encryption for a certain fixed key, dependent on this key?

DES, and any pure Feistel cipher, has even parity for any key. That's less clear for AES, and of no practical importance since parity of an arbitrary permutation can not be determined until $2^{128}-2$ correspondences are known, which is infeasible. That would however be a concern in the context of this comment, considering a reduced AES variant.
AES transformation can be viewed as a sequence of invertible transformations each processing only a small part of the state. All these transformations would be even, and so is the entire AES for any key (see also this question).