The curve 
$$(X-16)^3=XY\tag{1}\label{1}$$
is essential to Heegner's approach to the class number one problem for imaginary quadratic fields. We have the following “modular” parametrization 
\begin{equation}\tag{2}\label{2}(X,Y)=\left(2^{12}\Phi(\tau),j(\tau)\right),\end{equation}
where $\Phi(\tau)=\frac{\Delta(2\tau)}{\Delta(\tau)}$. Note that the function $\Phi(\tau)$ is a Hauptmodul for the group $\Gamma\_0(2)$.


My questions are:


1. The parametrization \eqref{2} can be deduced, quite laboriously, using Weierstrass elliptic functions and the product expansion for $\Delta$. However, when we clear the denominators, equation \eqref{1} (possibly) becomes an identity between modular forms and such identities should be easy to prove using the fact, that the space of modular forms has a finite dimension. Can this be done?
2. What can we say about $\Delta(2\tau)$? It is a modular form with respect to some group?



Modular parametrization of a curve of Heegner and Weber
Firstly, you have a typo.
The left side is $$(X-16)^3=-4096 + 3145728q - 729808896q^2+O(q^3)$$ while the right side is $$XY=4096 + 3145728q + 880803840q^2+O(q^3).$$


Looking at (19) of [Stark's paper](https://www.sciencedirect.com/science/article/pii/0022314X69900237), I think the relevant root of $(X-16)^3=Xj(\tau)$ is $X=-f\_2(\tau)^{24}$, which is $-2^{12}\Phi(\tau)$ in your notation.


Proceeding from this, you are then interested in a modular forms proof of
$$(2^{12}\Delta(q^2)+16\Delta(q))^3=2^{12}\Delta(q^2)\Delta(q)^2j(q).$$
I'm not sure there is anything better than noting that both sides are weight 36 modular forms of level 2, and then equating enough coefficients to exploit the finite dimensionality. Whether this is "easier" than the method you mention is not clear.


I don't know what you intend by (2). In general, $f(\tau)$ is a modular form on $\Gamma\_0(N)$, then $f(M\tau)$ is a modular form on $\Gamma\_0(MN)$. So clearly $\Delta(2\tau)$ is a modular form on $\Gamma\_0(2)$. Indeed, as you note, the quotient is a modular function on $\Gamma\_0(2)$.


