... and yet it clearly states

here that "The Riemann zeta function $\displaystyle \zeta(2n)$ may be computed analytically for even n using either contour integration

__or Parseval's theorem with the appropriate Fourier series__" (my underlining). Unfortunately, it doesn't tell you which Fourier series to use.

My guess is that there must be a function whose Fourier coefficients are $\displaystyle (-1)^n/n^3$. That would get round the problem of having to avoid anything that looks like the Apéry constant, yet still allows you to get $\displaystyle {\textstyle\sum} 1/n^6$ using Parseval's theorem. Have you tried the function $\displaystyle |x^3|$? (That's just a guess — I have no idea whether it will work.)

Actually $\displaystyle \pi^6/945$ (see the above link).