Let (A,D(A)) a diagonalizable generator of a C0−semigroup of contractions on a com- plex Hilbert space X, B ∈ L(C, Y ), Y being some suitable extrapolation space of X, and u ∈ L2(0,T;C). Under some assumptions on the sequence of eigenvalues Λ = {λk}k≥1 ⊂ C of (A, D(A)), we prove the existence of a minimal time T0 ∈ [0, ∞] depending on Bernstein's con- densation index of Λ and on B such that y′ = Ay+Bu is null-controllable at any time T > T0 and not null-controllable for T < T0. As a consequence, we solve controllability problems of various systems of coupled parabolic equations. In particular, new results on the boundary controllability of one-dimensional parabolic systems are derived. These seem to be difficult to achieve using other classical tools.