$R = \{(X, Y) \in \mathscr{P}(A)^2| X \subset Y \text{ and }X \neq Y \}$
I know that $(X,Y) \in R$ holds true since $X \subset Y$.
However I'm unsure if $(Y,X) \in R$ since if $Y \subset X$ then that would make $X = Y$ (I think) breaking the condition that $X \neq Y$. Am I correct in thinking this and concluding that this relation is not symmetric?
Yes, indeed, you are correct. The relation is not symmetric, and for the reason you post.
$$(X\subseteq Y\; \text{ AND }\;Y\subseteq X) \iff X=Y$$
Hence, since $(X, Y)\in R \implies (X\subseteq Y$ and $X\neq Y$), it cannot be the case that $Y \subseteq X$.