There are many such curves. For $p = 2$, one can just take two curves $E$ and $F$ with $E : y^{2} = f(x)$ and $F : y^{2} = g(x)$ where $f(x)$ and $g(x)$ define the same number field. Also, if $E$ and $F$ are isogenous, then the above statement is true for any $p$ (but the corresponding modular forms are the same then).
Rubin and Silverberg have a couple of papers (see the MathSciNet reference here) that handle $p = 3$ and $5$. In particular, for these $p$, given an elliptic curve $E/\mathbb{Q}$, they construct a one-parameter family of elliptic curves $E_{t}$ so that $E[p]$ and $E_{t}[p]$ are isomorphic as Galois modules. (This works because such $E_{t}$ are parametrized by a twist of $X(p)$, and $X(p)$ has genus zero for $p = 3$ and $5$).
For larger $p$, there are still pairs (but not quite as many), and the best place to look is in this paper of Tom Fisher. In particular, the largest known example is with $p = 17$, which I mentioned in my answer to the question here. This example is a pair of $17$-ordinary elliptic curves that have $\# E(\mathbb{F}_{\ell}) \equiv \# F(\mathbb{F}_{\ell}) \pmod{17}$ for all primes $\ell$.
