You say that Freudenthal does not elaborate his objections, but in a sense he did just that in his book Mathematics as an educational task (1973), Appendix I, starting: “The somewhat summary criticism I administered on several occasions to Piaget's work demands more detailed argument.” However, this addresses egregious mathematical misconceptions and methodological flaws (ill-chosen questions) more than the (“ontogeny parallels phylogeny”?) postulation you ask about.
Against that, his argument seems much shorter:
(p. 46): Bourbaki. How convincing this organization of mathematics is! So convincing that Piaget could rediscover Bourbaki's system in developmental psychology. Poor Piaget! He did not fare much better than Kant, who had barely consecrated Euclidean space as "a pure intuition" when non-Euclidean geometry was discovered! (...) Mathematics is never finished – anyone who worships a certain system of mathematics should take heed of this advice.
(p. 192): Impressed by Cantor's analysis he turned to studying the development of the number concept under this aspect. (...) Piaget believed that the concept of natural number could be entirely derived from potencies. (...) This may have seduced him to believe that it is also psychologically true; it was one of his ideas to trace in developmental psychology the system of mathematics he happened to be acquainted with.
Or, as reported by F. Goffree in The legacy of Hans Freudenthal (1993):
(p. 36): Piaget thought that the cognitive development in children took place from poor to rich structures whereas HF thought it was the other way round. Geometry showed something similar. In the Erlanger Program Klein had given a hierarchy of geometrical structures: topological, projective, affine and euclidean, in HF's terms from poor to rich. Children start by drawing irregular circles, anyone can see that. This was sufficient reason for Piaget to presume that their geometric development began with the topological structure. HF remarked condescendingly that those very same children were quite capable of distinguishing between correctly drawn circles and other figures.
(Note added: These last remarks, not sourced by Goffree, are in Revisiting mathematics education (1991), p. 27. So far as I can tell, the criticized “postulation” is most detailed in Épistémologie des mathématiques. Partie II. Chap. 8 & 11 (1961, transl. 1966).)