$A.\dfrac{x-y}{x+y}$是最简分式；

$B.\dfrac{y^{2-}x^{2}}{x+y}=y-x$，不是最简分式；

$C.\dfrac{x^{2}-y^{2}}{x^{2}y+xy^{2}}=\dfrac{\left(x+y\right)\left(x-y\right)}{xy\left(x+y\right)}=\dfrac{x-y}{xy}$，不是最简分式；

$D.\dfrac{x^{2}-y^{2}}{\left(x+y\right)^{2}}=\dfrac{x-y}{x+y}$，不是最简分式.

故选：$A$.