You can also do this "topologically". The idea is to separately calculate the (l-adic etale) Euler characteristics of the stack Y_0(p)$Y_0(p)$ in char. 0$\mathrm{char}$ $0$ and in char. p $\mathrm{char}$ $p$, then see how they must relate. Here we go:
Over C$\mathbb{C}$, and hence over Q_p-bar$\bar{\mathbb{Q}}_p$ as well, the Euler characteristic of Y_0(p)$Y_0(p)$ is (p+1)*(-1/12)$(p+1)\times(-1/12)$, since Y_0(p)$Y_0(p)$ is a (p+1)$(p+1)$-fold cover of Y=M_{ell}$Y=M_{ell}$.
On the other hand, over F_p-bar$\bar{\mathbb{F}}_p$, up to "homeomorphism" Y_0(p)$Y_0(p)$ is two copies of Y glued at the supersingular points, so the Euler characteristic is 2*(-1/12) - S$2\times(-1/12) - S$ (where S$S$ is the "number" of supersingular elliptic curves).
However, since Y_0(p)$Y_0(p)$ has semi-stable reduction (and is constant at infinity) over Z_p$\mathbb{Z}_p$, the special fiber is gotten from the generic fiber by contracting a bunch of "circles" to points, one for each nodal point of the special fiber; thus our second (char p$\mathrm{char}$ $p$) Euler characteristic is equal to our first (char 0$\mathrm{char}$ $0$) Euler characteristic plus S$S$. Comparing and solving for S$S$ gives the formula pretty quickly.
