But I have a very little knowledge about these things, but as far as my knowledge is concerned , there is some recent celebrated work by Manjul Bhargava and Arul Shankar, where in-fact they prove that a positive proportion of elliptic curves have rank $0$.
The have also used the recent results of Dokchitser, and proved it. So you can find their article here. To give a bird-eye view of what they have done, one can say that in fact, they are able to construct families such that exactly half of them have positive sign in their functional equation, and with average $3$-Selmer rank bounded by $7/6$. (This can be achieved by imposing appropriate conditions on the coefficients of the elliptic curve, and computing the root number as a product of local roots numbers.) Now work of the Dokchitser brothers on the parity conjecture implies that for elliptic curves with sign $+1$, the rank of the $3$-Selmer group is even. When combined with the bound of $7/6$, they deduce that the $3$-Selmer groups of the curves with sign $+1$ that lie in their family must be trivial.
Now (under some additional assumptions about the $3$-torsion, and some other technical assumptions, which they are able to impose on their family) by applying the results of Skinner and Urban on the Main Conjecture (which lets one pass from triviality of a Selmer group to non-vanishing of the $L$-function) they deduce that the curves in their family having sign $+1$ also have non-vanishing $L$-value at $s = 1$.
Now a positive proportion of elliptic curves overall lie in their family, and so putting all this together, one finds that a positive proportion of elliptic curves have both $3$-Selmer rank zero (and in particular, Mordell--Weil rank ( $g$ ) zero) and also analytic rank zero. Thus, a positive proportion of elliptic curves satisfy (the rank part of) BSD.
On the other hand, works by Kolyvagin and Don Zagier are also appreciable and gave rise to many theorems in this direction.
And regarding your parameterization, there is some parameterization done in the case of positive rank, so I can point the database maintained by Andrej, which is here, and it is well known that certain elliptic curves of the form $y^2= x^3-n^2x$ has a rank $0$, so there is a nice article by Keqin Feng and Maosheng Xiong which is here.
I am still searching for some nice articles, I will surely re-edit the post once I get them.