Consider a Riemannian manifold admitting a parallel (with respect to the Levi-Civita connection $\nabla^g$) symmetric bilinear form $\beta$ which is not a multiple of the metric $g$. Then, for an open set of constants $(a,b)\in\mathbb R^2$ $$g_{a,b}:=a g+b\beta$$ is a Riemannian metric, i.e., positive definite. The Levi-Civita connection of $g_{a,b}$ is $\nabla^g$. Moreover, the volume form of $g_{a,b}$ is a constant multiple (depending on $a,b$) of the volume form of $g$, as the volume forms $vol_{a,b}$ are parallel with respect to $\nabla^g$. Clearly, at $(a,b)=(1,0)$ we have $g_{a,b}=g$ and $$\frac{\partial vol_{a,b}}{\partial a}\neq0.$$ Hence, you always find a curve of Riemannian metrics with the same Levi-Civita connection and the same volume form under the above condition. Of course, explicit examples are provided by the answers of Ben and Liviu, but also by product metrics. Analogous arguments apply to the pseudo-Riemann case.

By taking the difference between two different metrics with the same Levi-Civita connection and the same volume form one immediately sees that the space of parallel symmetric bilinear forms must be at least two-dimensional.