what you've got here is a parabola rotated 45 degrees.

Probably the easiest way to proceed is to multiply your parameterized vector (x,y) by a rotation matrix of -45 degrees, solve for the directrix with the parabola in standard form, find a parameterized vector for the directrix line, then rotate that vector back 45 degrees.

A $\pm$45 degree rotation matrix is $\left(\begin{array}{cc}\cos(\pm 45deg) &\sin(\pm 45deg) \\ -\sin(\pm 45deg) &\cos(\pm 45deg)\end{array}\right)$