nt.number theory - Geometric meaning of fiber of modular parameterization over a point of an elliptic curve?

As Stankewicz explains, although elliptic curves appear in two guises in the modular parameterization $X_0(N) to E,$ first because $E$ is an elliptic curve, and secondly because $X_0(N)$ parameterizes elliptic curves, it is something of a red herring to think of these two appearances of elliptic curves as having anything to do with one another.

The reason that $X_0(N)$ appears in the problem of describing elliptic curves is because
elliptic curves have two dimensional $H^1$, and $X_0(N)$ is the Shimura variety over $mathbb Q$ associated to the group $GL_2$. Thus, as Stankewicz notes, Shimura curves (which are the Shimura varieties attached to twisted forms of $GL_2$) can equally well give parameterizations of elliptic curves.

Now the way we prove things about $X_0(N)$ (e.g. properties of its Heegner points, as in Pete Clark's answer) is using its moduli interpretation. But there are two things to bear in mind:

First, most (maybe all?) properties of the special points on $X_0(N)$, such as the Heegner
points, are special cases of general aspects of the theory of Shimura varieties (so although the proofs use the moduli interpretation, the statements can be formulated in a way that
doesn't refer to the moduli-theoretic interpretation, but instead refers to the interpretation
of $X_0(N)$ as a Shimura variety).

Second, the transfer of information is always from $X_0(N)$ to $E$. So while Heegner points give certain interesting points on $X_0(N)$ defined over class fields of quadratic imaginary fields, which can be mapped down to $E$ to give interesting points on $E$ defined over such fields, if one takes a random point on $E$ defined over a class field of an imaginary quadratic field and pulls it back to $X_0(N)$, it is not so easy to say what is going on with the preimages in general.

Finally, I think remark (3) in Pete Clark's answer is an interesting one. In the Mazur and Swinnerton-Dyer paper that David Hansen refer's to in his first comment, if I am remembering correctly, they also suggest that the images in $E$ of the critical points of the map from $X_0(N)$ to $E$ that lie on the geodesic arc joining $0$ to $infty$ in the upper half-plane may be worth studying. As with Birch's suggestion of Weierstrass points, I'm not sure how much has been done on this.

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