Let, $x(\theta,\varphi)=(R\sin \theta \cos \varphi, R\sin \theta\sin \varphi,R\cos \theta)$ be the usual parametrization of sphere. In general, we know that the first fundamental form of a surface $x=x(u,v)$, $y=y(u,v)$, $z=z(u,v)$, is given by:
$$(d\ell)^2=(x_u+y_u+z_u)(du)^2+(x_u x_v+y_u y_v+z_u z_v)(dudv)+(x_v+y_v+z_v)(dv)^2.$$
Also, the first fundamental form of sphere is given by:
$$(d \ell)^2=R^2(d\theta)^2+R^2\sin^2 \theta(d\varphi)^2.$$
So, we have that:
$$x_{\theta}^2+y_{\theta}^2+z_{\theta}^2=R^2$$
$$x_{\theta} x_{\varphi}+y_{\theta} y_{\varphi}+z_{\theta} z_{\varphi}=0$$
$$x_{\varphi}^2+y_{\varphi}^2+z_{\varphi}^2=R^2\sin^2 \theta$$
My question is: Is it possible to recover the parametrization from the equations above? Thanks in advance.
No. Indeed, you can have non-congruent (pieces of) surfaces with the same first fundamental form. Here's the simplest example that comes to mind:
Surface 1: $x(u,v) = (u,v,0)$ (the plane)
Surface 2: $x(u,v) = (\cos v,\sin v, u)$ (a cylinder)
Both have first fundamental form $du^2 + dv^2$.