Help on using the pumping lemma?
This language is regular. Since you're looking for equivalences mod 2 we need only consider the evenness or oddness of the count of the two kinds of symbols in the input. Consider a finite automaton with four states: EE, EO, OE, OO, where state EE records that the number of $a$'s seen so far is even and the number of $b$'s is also even. Similarly, being in state EO means you've seen an even number of $a$'s and an odd number of $b$s. States OE and OO are defined similarly.

Now consider the transitions between states. For example, in state EO with input a, the number of $a$'s changes from even to odd, while the number of $b$'s doesn't change, so we have the transition $\delta(\text{EO}, a) = \text{OO}$. The other seven transitions are defined similarly.

The start state is EE, since you've seen 0 $a$'s and 0 $b$'s and obviously 0 is an even number. The final states are EE and OO, for obvious reasons.