Let $\Omega$ be a domain with boundary $\partial\Omega$. Suppose I am given a weak formulation: $$b(u,v) = (f,v) \quad\forall v \in H^1(\Omega)$$ Assuming $b$ is nice enough, does the elliptic regularity result $$\lVert u \rVert_{H^2(\Omega)} \leq C\lVert f \rVert_{L^2(\Omega)}$$ follow just from the weak form without any explicit dependence on the boundary condition? So I mean do we care about the boundary condition to get this regularity result?
If so, I assume the same result will also hold for solutions to parabolic equations: so for $(u_t,v) +b(u,v) = (f,v)$ we have $$\lVert u(t) \rVert_{H^2(\Omega)} \leq C\lVert f(t) \rVert_{L^2(\Omega)}$$
If $b$ is "nice enough", for instance coercive, then elliptic regularity holds. Without giving flexibility to choose $b$ or the underlying function space, you are not leaving any freedom to choose the boundary condition, so your question is in sense trivial. In other words, the setting of the question fixes the boundary condition, so there is no "dependence" on the boundary condition.
In general, i.e., when $b$ and the function space are given, you need to have Agmon's condition that guarantees the bilinear form is coercive. Then you have regularity.
Even more generally, in the setting of general elliptic boundary value problems (that is when you don't necessarily rely on strong ellipticity), you need to check the Lopatisnky-Shapiro condition to get the standard elliptic theory working.