This can be shown easily by doing induction on your rel_list
hypothesis. Here's a generalized version of this that uses functions in the standard library:
Require Import Coq.Lists.List.
Section Lists.
Variables A1 A2 B1 B2 : Type.
Variables (RA : A1 -> A2 -> Prop) (RB : B1 -> B2 -> Prop).
Variables (f1 : A1 -> B1) (f2 : A2 -> B2).
Hypothesis parametric : forall a1 a2, RA a1 a2 -> RB (f1 a1) (f2 a2).
Lemma l : forall l1 l2, Forall2 RA l1 l2 ->
Forall2 RB (map f1 l1) (map f2 l2).
Proof.
intros.
induction H as [|a1 a2 l1 l2 HR H IH]; simpl; constructor; eauto.
Qed.
End Lists.