Parashat Balaq indirectly references "Tzippor" five times (by introducing Balaq as "ben Tzippor"). I don't believe Rashi brings down any background regarding Tzippor. If the Torah refers to Balaq as "ben Tzippor" so often, who was this mysterious figure?
The Zohar HaQadosh (3:184) writes:
בן צפור כמו שאמרו. אבל בן צפור ממש שהרי כשפיו היו בכמה מינים של אותה צפור
[Balaq was called] "son of Tzippor" as was already stated; but, [he was] truly a son of birds for his sorcery was conducted with a few species of the same bird!
The Ohr Hachayim has a cool explanation of the phrase בן ציפור - that Balak, who (he says) was a greater magician than Bilaam, was an expert ornithomancer, and had a magical bird that told him everything that he needed to know. In context of the first pasuk in the parsha, "Balak saw, through his bird, everything that the Jewish people did...", and via that same magic bird, he saw that Midian was next.
Perhaps, therefore, Balak is often referred to in this manner as a way of mocking him or the like, pointing out his reliance on a bird (while the Jewish people were relying on Hashem). There are many other reasons I could think of to explain why the Torah would want to emphasize this feature of Balak.