Matroid isomorphism

Question

I am working on some generalization of the paper Projective orientations of matroids by Gel'fand, Rybnikov and Stone to the more general context of matroids over hyperfields.

There is a technical detail in their proof of Theorem 1 (page 133) that I am not able to understand.

Question Let $M$ be a matroid with $\operatorname{rk}(M)\leq2$ and such that there are no coparallel elements in it. Why can we deduce that $M$ is isomorphic to $U_{2}(4)$, the rank $2$ uniform matroid over $4$ elements?

Answer 1

The authors are requiring more than just the sentence in question. The whole paragraph is needed. The lemmata proven say a minimal counter example $M$ must have no parallel elements and corank at most 2. The authors then note this also holds when replacing $M$ with its dual. So, the claim really is that if $M$ is connected, does not have any parallel nor any coparallel elements, and both the rank and corank are at most 2. Then $M$ is isomorphic to $U_2(4)$.