Let a pair $(A,B)$ of bounded linear operators acting on a Hilbert space be a solution of the operator equations $ABA=A^2$ and $BAB=B^2$. When $A$ is a paranormal operator, we explore some behaviors of the operators $AB$, $BA$, and $B$. In particular, if $A$ or $A^*$ is a polynomial root of paranormal operators, we show that Weyl type theorems are satisfied for the operators $AB$, $BA$, and $B$.