See [this article][1] (Hebrew).

The *piyutim* (*shivatot*) classically recited by *nusach* Ashkenaz at *musaf* of *shekalim* and *hachodesh* (although now omitted by many, many congregations) were composed by the Kalir.

The author notes that the Kalir actually composed many *shivatot* for all of the *arba parshiyot*, but he claims that for some reason, only two (for *shekalim* and *hachodesh*) had arrived in Europe by the 11th century, where they were recited at *musaf*.

He proves this from the fact that R. Meir Shatz of Worms felt the need to compose *shivatot* (in the style of the Kalir) for the two missing *shabbatot*. However, reciting these *shivatot* never became accepted as mainstream *minhag* Ashkenaz.

What is even more surprising is that some of the Kalir's *shivatot* for *zachor* and *parah* did eventually make it into Ashkenaz in the 12th century (they appear in Ashkenazi manuscripts and are the subjects of Ashkenazi commentaries) but they were never adopted as part of *minhag* Ashkenaz.

Bottom line, the answer is that there are no *piyutim* recited at *musaf* of *zachor* and *parah* because the Kalir's *piyutim* were not available in Ashkenaz when *minhag* Ashkenaz was getting started, and later developments (R. Meir Shatz's composition or the subsequent arrival of the Kalir's *piyutim*) were not sufficient to change the status quo.

  [1]: https://web.archive.org/web/20160422082022/http://shituf.piyut.org.il/story/1384