  The parity of Selmer ranks for elliptic curves defined over the rational
numbers $\mathbb{Q}$ with good ordinary reduction at an odd prime $p$ has been
studied by Shekhar. The proof of Shekhar relies on proving a parity result for
the $\lambda$-invariants of Selmer groups over the cyclotomic
$\mathbb{Z}_p$-extension $\mathbb{Q}_\infty$ of $\mathbb{Q}$.
  This has been further generalized for elliptic curves with supersingular
reduction at $p$ by Hatley and for modular forms by Hatley--Lei. In this paper,
we prove a parity result for the $\lambda$-invariants of Selmer groups over
$\mathbb{Q}_\infty$ for the symmetric square representations associated to two
modular forms with congruent residual Galois representations. We treat both the
ordinary and the non-ordinary cases.
