Parallel-Support Rank and a One-Sided Merino--Welsh Inequality

For a finite loopless, coloopless matroid M, let P(M) be the union of its nontrivial parallel classes. We prove that T_M(0,2) is at least T_M(1,1) whenever the rank of P(M) is at least r(M)-1. The inequality is strict at parallel-support corank one, and equality in the full stated class occurs exactly for direct sums of copies of U_{1,2}. We also give the dual series-class statement, a graphic interpretation, and connected and disconnected counterexamples showing that the rank threshold is sharp without further hypotheses. Status: Public Beta; internally verified candidate proof; external mathematical review pending.