My experience is that the Pastafarian religion is indeed a parody religion. In contrast, TST has all of the indicia of a sincere religion – you can look here for a summary of their beliefs. Their third tenet is "One's body is inviolable, subject to one’s own will alone", from which a woman's right to chose to have an abortion would flow.
So what remains to be decided is whether this is "a religion", or is it a philosophical system. The Supreme Court has not and in principle could not "define religion", since the point of the First Amendment is that the government may not consign Hinduism or Taoism to some lesser status because these religions do not have a single deity. In Torcaso v. Watkins, 367 U.S. 488 the court footnoted the observation that "Among religions in this country which do not teach what would generally be considered a belief in the existence of God are Buddhism, Taoism, Ethical Culture, Secular Humanism and others". Some decisions (US v. Seeger, Welsh v. US), both regarding religious exemptions to the draft found (from Welsh) that the excemptin clause
contravenes the Establishment Clause of the First Amendment by
exempting those whose conscientious objection claims are founded on a
theistic belief, while not exempting those whose claims are based on a
secular belief. To comport with that clause, an exemption must be
"neutral" and include those whose belief emanates from a purely moral,
ethical, or philosophical source
and furthermore, the exemption was not found unconstitutional, it was broadened in scope to go beyond theistic belief systems. Thus
the test of religious belief under § 6(j) is whether it is a sincere
and meaningful belief occupying in the life of its possessor a place
parallel to that filled by the God of those admittedly qualified for
the exemption.
There has been some back-pedaling in Wisconsin v. Yoder, which posited that
Although a determination of what is a "religious" belief or practice
entitled to constitutional protection may present a most delicate
question, the very concept of ordered liberty precludes allowing every
person to make his own standards on matters of conduct in which
society as a whole has important interests. Thus, if the Amish
asserted their claims because of their subjective evaluation and
rejection of the contemporary secular values accepted by the majority,
much as Thoreau rejected the social values of his time and isolated
himself at Walden Pond, their claims would not rest on a religious
basis. Thoreau's choice was philosophical and personal, rather than
religious, and such belief does not rise to the demands of the
Religion Clauses
and in Thomas v. Review Bd. it is asserted that "Only beliefs rooted in religion are protected by the Free Exercise Clause, which, by its terms, gives special protection to the exercise of religion". But note that this distinguishes personal philosophical beliefs from religious beliefs, and does not say what the necessary or sufficient conditions are for being deemed a religion.
TST does indeed purport to be a religion and not simply a philosophical system. As far as I know, SCOTUS has not, at least in the past 100 years, ruled that any purported religion is not in fact a religion.