Roman Catholic Churches (about 25% of these parishes, using the Liturgy of St. Gregory, similar to the Tridentine Mass).
By far the largest group of these parishes in North America is represented by the Western Rite Vicariate of the North American Archdiocese of the Antiochian Orthodox Church, which has neither autocephaly (complete hierarchical independence) nor autonomy (governance of internal affairs, but its primate is appointed by and answerable to a parent jurisdiction's synod), but reports ultimately to the Patriarch of Antioch. The Russian Orthodox Church Outside Russia (ROCOR) also has a small number of Western-Rite parishes.
One can compare the situation of Western Orthodox parishes with the analogous status of autonomous Eastern-Rite Catholic churches. For centuries, there have been hierarchical churches in full communion with the Vatican, but which the Pope allows to follow customs and rules like those of the Eastern Orthodox Church, (e.g., they confirm newly baptized infants via chrismation, they have married priests, their churches have iconostases, etc.).
The present-day influence of the OrthodoxChurch encompasses the territories associated with the former Byzantine and Russian empires: Eastern Europe, Asia (Russia/Siberia), and parts of the Middle East and Africa.
In the Theology of the OrthodoxChurch it is most important to understand that Christ, from the moment of conception was 100% God and 100% man. Therefore it is correct to say that Mary is indeed, the Theotokos, the Birth-giver of God, and that she is the greatest of all humans ever to have lived.
The Antiochian OrthodoxChurch, The Russian OrthodoxChurch Outside Russia, and the Holy Orthodox Catholic and Apostolic Church of America (formerly connected with the Vicar Bishop of the (Western) OrthodoxChurch of France-ECOF), all have Western Rite parishes.