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 
$$\sup_{x\in X}\,\inf_{\rule{0pt}{6.6pt}y\in Y}\,f(x,y)
=\inf_{\rule{0pt}{7pt}y\in Y}\,\sup_{x\in X}\,f(x,y)$$
was given in [this paper][1]; you may also want to see the corresponding [zbMATH review][2].

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. 


  [1]: https://link.springer.com/article/10.1007/BF01158303
  [2]: https://www.zbmath.org/?q=au%3Apinelis%20ti%3Adeterminacy