If the underlying field is $\mathbb{C}$, there is a bijective map between a given elliptic curve and $\mathbb{C} \mathbin{/} \Lambda$, where $\Lambda$ is a lattice uniquely determined by the elliptic curve. This map is a holomorphic isomorphism and a group isomorphism as well, and is as such unique. In $\mathbb{C} \mathbin{/} \Lambda$, the group law is merely the addition.
I think that the above is correct, but please let me know if some finesse escapes me (regarding the uniqueness of the map and of the lattice, for instance).
The proper question question is: Is there a similar parameterization for the case that the underlying field is a finite field (some analogon to Weierstrass's $\wp$-function)? That is, does a parameterization space exist, in which the group law is an addition (or some kind of elementary operation)? Or does otherwise the group law just happen to work in analogy to the complex case, but does not have deeper connections, as in the complex case?
The same question holds also for bilinear pairings (in particular for the Weil-pairing).