We consider several coupled systems of one-dimensional linear parabolic equations where only one equation is controlled with a distributed control. For these systems we study the minimal null-control time that is the minimal time needed to drive any initial condition to zero. In a previous work [Comptes Rendus. Mathématique, Tome 361 (2023)] we extended the block moment method to obtain a complete characterization of the minimal null-control time in an abstract setting encompassing such non-scalar controls. In this paper, we push forward the application of this general approach to some classes of 1D parabolic systems with distributed controls whose analysis is out of reach by the usual approaches in the literature like Carleman-based methods, fictitious control and algebraic resolubility, or standard moment method. To achieve this goal, we need to prove refined spectral estimates for Sturm-Liouville operators that have their own interest.