Another approach I got from the paper *Fibered Links in $S^3$*, available at https://doi.org/10.1016/j.exmath.2016.06.006.

Torus knots are algebraic, so they are fibered. It is known that the fiber surface of a fibered knot is the minimal genus Seifert surface. Example 3.2 of the aforementioned paper presents a fiber surface, hence the min genus Seifert surface, for the torus knot $T(p,q)$ as a blackboard framed embedding of the complete bipartite graph $K_{p,q}$ under a certain embedding in $S^3$. Here it is in the case $(p,q)=(3,4)$, Figure 6 from the paper.

The Euler characteristic of such a surface is just $(p+q)-(pq)$. On the other hand, the surface has 1 boundary component so its Euler characteristic is $1-2g$ where $g$ is the genus. Thus $T(p,q)$ has genus $(p-1)(q-1)/2$.