Is the parabolic heat equation with pure neumann conditions well posed?

The parabolic heat equation is a partial differential equation given by $\frac{du}{dt}=\nabla^2u+f$. If i impose an initial condition u(x,0) and pure homogeneous neumann boundary conditions that satisfy the compatibility conditions with respect to the source term f(x), does this result in a well posed problem. That is, does a unique solution exist?

I know that if we consider the steady state heat equation $-\nabla^2u=f$, pure neumann conditions imply that a solution exists, but the solution is not unique without imposing additional constraints. Of course, the steady state heat equation is an elliptic pde, so i'm not sure if the same can be said about the parabolic heat equation pde as well.

Yes, it is well-posed (provided $f$ and the initial data are suitable, at least). There is a nice physical interpretation of the difference between the two situations.
I assume you're working on a nice bounded domain with piecewise smooth boundary. For the elliptic problem $-\nabla^{2}u = f$, the difference of two solutions $u_1,u_2$ would have to be a constant because $w = u_1-u_2$ would satisfy $\nabla^{2}w =0$ and $\frac{\partial w}{\partial n}$, which gives $$0=\int_{\Omega}w\nabla^{2}u dV = \int_{\Omega}\nabla\cdot(w\nabla w)-|\nabla w|^{2}dV =-\int_{\Omega}|\nabla w|^{2}dV.$$
For the parabolic problem, the difference $w$ of two solutions would satisfy $$\frac{d}{dt}\frac{1}{2}\int_{\Omega}w^{2}dV=\int_{\Omega}w\nabla^{2}wdV=-\int_{\Omega}|\nabla w|^{2}dV \le 0 \\ \implies 0\le\int_{\Omega}w^{2}(x,t)dV \le \int_{\Omega}w^{2}(x,0)dV=0.$$ Interesting difference.