The CFL condition states that the "mathematical domain of dependence" must be (asymptotically) contained in the numerical domain of dependence. For hyperbolic problems, this provides a bound $\Delta t < C \Delta x$ that is useful at all resolutions. For a parabolic problem, it merely requires that $\Delta t \in o(\Delta x)$ in the limit $\Delta x \to 0$. That is, $\Delta t$ must go to zero strictly faster than $\Delta x$. You cannot conclude based purely on CFL theory that the time step must indeed go to zero at least as fast as $(\Delta x)^2$. This result is readily established using von Neumann stability analysis.
I recommend Chapter 4 of Trefethen's Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations for further details on this subject.