Here is a (probably non-optimal) statement that may apply in your situation. In your situation with curves, the hypothesis says that you need $X$ and $Y$ to be Gorenstein.

We start by defining the $G_r$ condition.

**Definition** [[Reiten and Fossum 1970][1], p. 142]**.** Let $X$ be a locally Noetherian scheme and fix an integer $r \ge 0$. We say that $X$ *satisfies $G_r$* if $\mathcal{O}_{X,x}$ is Gorenstein for every point $x \in X$ such that $\dim(\mathcal{O}_{X,x}) \le r$.

**Claim.** *Let $X$ and $Y$ be Noetherian schemes satisfying $G_1$ and $S_2$. Suppose that $f\colon X \to Y$ is a finite surjective morphism. If $\mathscr{F}$ is a coherent reflexive sheaf on $X$, then $f_*\mathscr{F}$ is a coherent reflexive sheaf on $Y$.*

*Proof.* On Noetherian schemes satisfying $G_1$ and $S_2$, reflexivity is equivalent to being $S_2$ (in [Samuel][2]'s [[1964][2], Proposition 6] sense [[Vasconcelos 1968][3], Definition 1.1]) by [[Vasconcelos 1968][3], Theorem 1.4; [Hartshorne 1994][4], Theorem 1.9]. The claim then follows since the $S_r$ property is preserved under pushforward by finite surjective morphisms by [[EGAIV$_2$][5], Proposition 5.7.9]. $\blacksquare$

I wanted to prove a statement for non-finite morphisms as well, and for integral schemes, you can say a bit more:

**Claim.** *Let $X$ and $Y$ be integral Noetherian schemes satisfying $G_1$ and $S_2$. Suppose that $f\colon X \to Y$ is a proper dominant morphism with all fibers of the same dimension. If $\mathscr{F}$ is a coherent reflexive sheaf on $X$, then $f_*\mathscr{F}$ is a coherent reflexive sheaf on $Y$.*

*Proof.* The fact that $f_*\mathscr{F}$ is coherent and normal follows from the proof of [[Hartshorne 1980][6], Corollary 1.7]. By [[Hartshorne 1994][4], Remark 1.11], to show $f_*\mathscr{F}$ is reflexive, it therefore suffices to show that it satisfies $S_1$. But being $S_1$ is equivalent to torsion-freeness for integral noetherian schemes [[Hartshorne 1994][4], Lemma 1.5], hence the claim follows by the fact that torsion-freeness is preserved under pushforwards by dominant morphisms. $\blacksquare$

*Remark.* There are other notions called $G_r$ in the literature. For example, $S_r$ + Reiten and Fossum's condition $G_r$ is what Marinari calls $G_r$ in [[Marinari 1972][7], Definition 4.5]. The condition $S_{r+1} + G_r$ is what Ischebeck calls $G_r$ in [[Ischebeck 1969][8], Definition 3.16].

**Edit 1.** Added the hypothesis that $f$ is surjective in the first claim.

**Edit 2.** To address [Shrugs][9]'s request in the comments, I added the definition of $G_r$ with some references to other definitions in the literature.

  [1]: https://mathscinet.ams.org/mathscinet-getitem?mr=289503
  [2]: https://doi.org/10.24033/bsmf.1608
  [3]: https://doi.org/10.1090/S0002-9939-1968-0237480-2
  [4]: https://doi.org/10.1007/BF00960866
  [5]: https://doi.org/10.1007/bf02684322
  [6]: https://doi.org/10.1007/BF01467074
  [7]: http://www.numdam.org/item/RSMUP_1972__48__67_0/
  [8]: https://doi.org/10.1016/0021-8693(69)90090-8
  [9]: https://mathoverflow.net/users/127554/shrugs