See this article (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.

See this article (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.

See this article (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.

See this article (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.