Büchi-like characterizations for Parikh-recognizable omega-languages

B\"uchi's theorem states that $\omega$-regular languages are characterized as languages of the form $\bigcupi Ui Vi^\omega$, where $Ui$ and $Vi$ 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 $Ui Vi^\omega$, where $Ui$ and $Vi$ are Parikh-recognizable. Furthermore, we show that the class of such languages, where $Ui$ is Parikh-recognizable and $Vi$ 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 $Ui$ and Parikh-recognizable $V_i$.