  Let $G$ be a graph and let $g, f$ be nonnegative integer-valued functions
defined on $V(G)$ such that $g(v) \le f(v)$ and $g(v) \equiv f(v) \pmod{2}$ for
all $v \in V(G)$. A $(g,f)$-parity factor of $G$ is a spanning subgraph $H$
such that for each vertex $v \in V(G)$, $g(v) \le d_H(v) \le f(v)$ and
$f(v)\equiv d_H(v) \pmod{2}$. We prove sharp upper bounds for certain
eigenvalues in an $h$-edge-connected graph $G$ with given minimum degree to
guarantee the existence of a $(g,f)$-parity factor; we provide graphs showing
that the bounds are optimal. This result extends the recent one of the second
author (2022), extending the one of Gu (2014), Lu (2010), Bollb{\'a}s, Saito,
and Wormald (1985), and Gallai (1950).
