When is $\mathfrak{S}_n \times \mathfrak{S}_m$ a subgroup of $\mathfrak{S}_p$?

Assume that $m+n>p$. Then at least one of $m,n$ - say $n$ - satisfies $n>p/2$. For $n \ge 7$, the only faithful transitive action of $S_n$ of degree less than $2n$ is the natural one. (See, for example,

Liebeck, Martin W.; Praeger, Cheryl E.; Saxl, Jan. A classification of the maximal subgroups of the finite alternating and symmetric groups. J. Algebra 111 (1987), no. 2, 365–383.)

So if $m+n \ge 13$, then there is a set of $n$ points on which $S_n$ acts naturally. Then the centralizer of $S_n$ in $S_p$ must fix every point in this orbit of $S_n$, so it has order at most $(n-p)!$, and hence it cannot contain $S_m$.