The office alleged that the parents gave their 1-year-old son to a "third party" during the demonstration on July 27, claiming this "endangered the boy's health and welfare, as well as causing him physical and moral harm."
The statement added that prosecutors would apply to a court to relieve the parents of their parental rights.
Thousands turned out on July 27 and August 3 for unsanctioned rallies in Moscow to protest a move by authorities to bar opposition candidates from taking part in municipal elections next month.
Authorities said at the time that the candidates were barred from running because they had failed to obtain a sufficient number of signatures to be put on the ballot paper. Opposition politicians and supporters have rejected this allegation.
A total of over 2,000 people were arrested in the two demonstrations, which were peaceful but broken up by well-equipped riot police.