How do I parameterize the intersection of x 2 − y 2 = 11 and...

Use $\sec$ and $\tan$, or $\cosh$ and $\sinh$ instead of $\cos$ and $\sin$. You just need to remember that $1+{\tan }^{2}t={\sec }^{2}t$, or that ${\cosh }^{2}t-{\sinh }^{2}t=1$. I'll go with the hyperbolic functions. Since $x^2-y^2=11$ is equivalent to $(x/\sqrt{11}{)}^2-(y/\sqrt{11}{)}^2=1$, we may simply let $x=\sqrt{11}\cosh t$ and $y=\sqrt{11}\sinh t$. The second equation immediately gives z. One possible parametrization is
$t\mapsto (\sqrt{11}\cosh t,\sqrt{11}\sinh t,(11/2)\sinh (2t)),$
for example. Observe that the intersection is not connected, and the other half is parametrized by
$t\mapsto (-\sqrt{11}\cosh t,\sqrt{11}\sinh t,-(11/2)\sinh (2t)).$