In 2010, Masser and Zannier proved that there are at most finitely many complex numbers t, not equaling 0 or 1, such that the two points on the Legendre elliptic curve y^2=x(x-1)(x-t) with x-coordinates 2 and 3 are simultaneously torsion. Recently, Stoll proved that there is in fact no such t, and it is his result that inspires our work. In this talk we will focus on the Weierstrass family of elliptic curves E_t: y^2=x^3+t, and show that in many instances there will be no parameter t such that the points (a,*) and (b,*) are simultaneously torsion in E_t. In contrast to the original approach of Masser and Zannier, our approach is dynamical. We focus on studying whether a and b are simultaneously preperiodic for a Latt\`es map.