In this paper, we study some new connections between parabolic Liouville-type theorems
and local and global properties of nonnegative classical solutions to superlinear parabolic
problems, with or without boundary conditions. Namely, we develop a general method
for derivation of universal,  pointwise a~priori estimates of solutions from Liouville-type
theorems, which unifies and improves many results concerning a~priori bounds, decay
estimates and initial and final blow-up rates. For example, for the equation $u_t-\Delta u=u^p$
on a domain $\Omega$, possibly unbounded and not necessarily convex, we obtain initial and
final blow-up rate estimates of the form
$u(x,t)\leq C(\Omega,p)\,(1+t^{-\frac{1}{p-1}}+(T-t)^{-\frac{1}{p-1}})$.
Our method is based on rescaling arguments combined with a key ``doubling'' property, and
it is facilitated by  parabolic Liouville-type theorems for the whole space or the half-space.
As an application of our universal estimates, we prove a nonuniqueness result for an initial
boundary value problem.