Let a,b∈Q¯ be such that exactly one of a and b is an algebraic integer, and let ft(z)=z2+t be a family of quadratic polynomials parametrized by t∈Q¯. We prove that the set of all t∈Q¯ for which there exist m,n≥0 such that fmt(a)=fnt(b) has bounded height. This is a special case of a more general result supporting a new bounded height conjecture in arithmetic dynamics. This is joint work with DeMarco, Ghioca, Krieger, Tucker, and Ye.