Expanding stand-alone simulation-based proofs to UC proofs

This is a follow-up question to Mikero’s answer to Simulation-based proofs and universal composability proofs.

Let there be some protocol $\pi$ running between two parties $A$ and $B$. Furthermore, assume that I have proven $\pi$ secure using a stand-alone simulation-based proof. That is, I have written some proof $\mathsf{P}$, constructing a simulator for $A$ (and later for $B$) in the ideal world that interfaces with adversary $\mathcal{A}$ in a real-world execution of $\pi$ where it pretends to be $B$.

This results in sequential composability. Now, I would additionally like to prove security under parallel composition, but avoid the increased complexity of a full UC proof.

I understand that one can assume adversary $\mathcal{A}$ to be a dummy adversary (see the answer in that previous question, a combination with the external environment). If the simulators in my proof $\mathsf{P}$ do not make use of rewinding arguments and instead run and interfere with $\mathcal{A}$ straightline, is this a valid argument to show security under parallel composition?