We give birational parametrizations of pairs (C,T) where T is a 5-torsion point on the Jacobian of a genus-2 curve C, possibly satisfying one or more of the following additional conditions: T = (P) - (P_0) for some points P_0, P in C with P_0 Weierstrass; C has one or more additional rational Weierstrass points; Jac(C) has real multiplication by (1+\sqrt{5})/2, and T is a \sqrt{5}-torsion point.