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 $\Delta x$, which causes information to propagate infinitely fast, thus matching the mathematical (continuum) behavior. 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.
Improper usage: The term "CFL" is sometimes misused to refer to whatever is the appropriate sharp stability requirement for an explicit method applied to the problem being considered. Indeed, CFL analysis is too weak to provide $\Delta t \sim\Delta x^2$ for parabolic problems or exotic dispersive wave problems (such as VLF Whistler waves).