Using OddQ and EvenQ in piecewise functions: Buggy?

As suggested by J.M., the solution is to use Mod:

f[k_] := Piecewise[{{1, k==0}, {x^k/k!, Mod[k, 2]==1}, {x^k/k!, Mod[k, 2]==0}}]

This returns the expected values for both

Sum[f[k], {k,0,5}]

and for

Sum[f[k], {k, 0, Infinity}]

As an interface-design issue, the choice to make *Q functions return a value even for generic symbols seems like a very poor and inconsistent one. But at least now I know to be wary...