In the first part of the article, we establish a necessary and sufficient condition ensuring classical solvability of the Cauchy problem with zero initial data for uniformly parabolic equations whose coefficients are Hölder continuous and whose right-hand sides possess a local continuity modulus. In the second part, we find a representation for a classical solution provided that the latter exists. Herewith, the growth of the right-hand side of an equation is arbitrary as $t\to 0$ and preassigned as $|x|\to\infty$. In the last part, we obtain necessary and sufficient conditions for classical solvability of the Cauchy problem with zero initial data for parabolic equations with constant coefficients and right-hand sides infinitely differentiable for $t>0$.