Does this DFA have a solution?
This is doable with a DFA. Hint: We need to keep track of three things simulataneously:


The parity of a's (have we seen an odd or even number of a's so far?)
The parity of b's
The parity of c's


So there are 8 possibilities for what we've seen so far:


Even number of a's, even number of b's, even number of c's.
Odd number of a's, even number of b's, even number of c's.
Even number of a's, odd number of b's, even number of c's.
...


Does that help?