Two different definitions of cross-correlation
Start with:

$$ R_{XY}(\tau) = \int_{-\infty}^{+\infty} x(t) y(t+ \tau) dt $$

Then it follows:

$$ R_{YX}(\tau) = \int_{-\infty}^{+\infty} y(t) x(t+ \tau) dt $$

Let $s = t+ \tau $:

$$ R_{YX}(\tau) = \int_{-\infty}^{+\infty} y(s -\tau ) x(s ) ds $$

Reverse the product and substitute $t$ for $s$:

$$ R_{YX}(\tau) = \int_{-\infty}^{+\infty}  x(t ) y(t -\tau ) dt $$

Now it is pretty clear that

$$ R_{XY}(\tau) = R_{YX}(-\tau) $$ 

The question of odd or even is irrelevant because they are not the same function.

Note:  Your second function is assumed to be $ R_{YX}(\tau) $, otherwise your are simply giving contradictory definitions to the same function.