  B\"uchi's theorem states that $\omega$-regular languages are characterized as
languages of the form $\bigcup_i U_i V_i^\omega$, where $U_i$ and $V_i$ are
regular languages. Parikh automata are automata on finite words whose
transitions are equipped with vectors of positive integers, whose sum can be
tested for membership in a given semi-linear set. We give an intuitive automata
theoretic characterization of languages of the form $U_i V_i^\omega$, where
$U_i$ and $V_i$ are Parikh-recognizable. Furthermore, we show that the class of
such languages, where $U_i$ is Parikh-recognizable and $V_i$ is regular is
exactly captured by a model proposed by Klaedtke and Ruess [Automata, Languages
and Programming, 2003], which again is equivalent to (a small modification of)
reachability Parikh automata introduced by Guha et al. [FSTTCS, 2022]. We
finish this study by introducing a model that captures exactly such languages
for regular $U_i$ and Parikh-recognizable $V_i$.
