And more generally, actually:
Let $M \subset \mathbb{R}^3$ be a regular surface locally isometric to the plane and suppose a regular parametrization of $M$ is $\phi: \mathbb{R}^2 \to M$ given by $(u, v) \mapsto (x(u, v), y(u, v), z(u, v))$, that sends every point in the plane to a point of $M$. Define another map $Q: \mathbb{R}^2 \to M $ in the following manner:
$$\mathbb{R}^2 \ni p \mapsto \phi(A(p)) $$
where $A(p)$ is an arbitrary isometry of $\mathbb{R}^2$ (and also $\phi$ is chosen carefully enough that it has the same coefficients of the fundamental form as the plane). Is it true that $Q$ is an isometry of $M$? If so, can the reuslt be generalized to "images of isometries under regular parametrizations are themselves isometries"? And again, if that's true, how would I state this for riemannian manifolds in general?
The question has been answered in the comments.