how to parameterize the boundary of a a parabolic cylinder

Question

Let $\Sigma$ be the surface of equation $x^2+y+z=1$ with $y,z \geq 0$. Let $F(x,y,z)=(6y,1-z^2,3x)$ be a vector field. I have to compute the circuitry of $F$ in $\gamma=\partial \Sigma$ without the Kelvin- Stokes theorem. I know that $\Sigma$ is parabolic cylinder but I don’t know how to parameterize $\partial \Sigma$. Some ideas?