How to express the surface $ \ y^{2}+z^{2}=15 \ $ between $x=-6$ and $x=8$ in parametric form?
From $ y^{2}+z^{2}=15 $ , we can have $$ y=\sqrt {15} \cos t \quad \text{ and }\quad z=\sqrt{15} \sin t,$$ but how to manage $x=-6$ to $x=8$? Is it like $-6 \leq x \leq 8$? Any help is appreciated.
Your surface is a cylinder symmetric about the $x$-axis. You need two parameters for a 2-dimensional surface. A "natural" choice of the two parameters would be
(1) the $x$ coordinate (this brings you down to a circle), $x\in[-6,8]$, and
(2) the angle $\theta$ (or $t$ if you prefer) specifying a point on the circle and ranging from $0$ to $2\pi$.
Your attempted solution seems right on. Every point on the surface can be written as $$ (x,y,z) = (x,\sqrt{15}\cos\theta,\sqrt{15}\sin\theta). $$
If necessary, we can bring the parameters into the range $0$ to $1$ using a linear transformation; but your problem statement does not specifically ask you to have parameters in the $[0,1]$ range (or does it?)