Let $f\colon X\times Y\to\mathbb R$, where $X$ and $Y$ are any sets. Suppose that the function $f$ is generalized concave-convex in the sense that for any $x_0,x_1$ in $X$, any $y_0,y_1$ in $Y$, and any $t\in[0,1]$ there exist $x_t\in X$ and $y_t\in Y$ such that for all $x\in X$ and $y\in Y$ $$f(x_t,y)\ge(1-t)f(x_0,y)+tf(x_1,y)$$ and $$f(x,y_t)\le(1-t)f(x,y_0)+tf(x,y_1).$$
A necessary and sufficient condition for $$\sup_{x\in X}\inf_{y\in Y}f(x,y)=\inf_{y\in Y}\sup_{x\in X}f(x,y)$$ was given in this paper; you may also want to see the corresponding zbMATH review.
In Parthasarathy's minimax theorem, the "payoff" function $f$ is affine in each of its two arguments and hence concave-convex, and hence generalized concave-convex.