The equations $X^4-2Y^4 = \pm Z^2$ have a long and storied history, dating back to at least 1643 (Fermat).
It is a classical result — easy to replicate — that given any integer solution with $x^2>1$, another solution with $1 \le x' < x$ can be derived. Hence by reversing the process and starting with $x=y=z=1$, all integer solutions can eventually be derived.
However, I’m interested in whether or not it would be possible to develop an integer parameterization (along the lines of the classical solution to the Pythagorean equation $A^2+B^2=C^2$). Has this already been done? If not, is there anything (such as the presence of a fourth power) that precludes the possibility of developing a complete solution in the form of an integer parameterization?