Let a, b ∈ $Q¯$ be such that exactly one of a and b is an algebraic integer, and let f(z) := z$\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}$ + t be a family of polynomials parameterized by t ∈ $Q¯$. We prove that the set of all t ∈ $Q¯$ for which there exist m, n ≥ 0 such that f(a) = f(b) has bounded height. This is a special case of a more general result supporting a new bounded height conjecture in arithmetic dynamics.