Much to the delight of my colleagues and Hungary Exchange Facebook group, many Hungarian church records have become digitized and freely viewable online at FamilySearch.org. You can now browse and enjoy these records at home, instead of needing to frequent your local FHC, and you will spend less money from having to order the microfilm. The Reformed and Roman Catholic baptism indexes, as well as the Jewish Vital Records indexes, don't appear to link up with their proper digitized image yet, so there certainly is still work to be done behind the scenes.
The best way to find your parish is to search the microfilm catalog, under the Place category, at FamilySearch.org/catalog/search. Most localities will come up this way, however not all villages or towns had a church or synagogue for each religion - hence requiring the citizens to travel to the nearest neighboring village. There are several gazetteers that help guide you to the correct parishes and many of them are now online. Here you can find the 1877, 1888 and 1893 gazeetters. The Hungarian National Archive's church record microfilm catalog, which was edited, compiled and prepared by Dr. Kálniczky László in 1998, is a gazetteer which may reflect earlier parishes. You can find here Volume 1 and Volume 2.
Once you find the parish you need and pull up the microfilm catalog title, you will be able to determine if the content is digitized and available. On the catalog title, under Format and next to the microfilm number, you will find a magnifying glass icon (indicating the microfilm is [partially] indexed) and a camera icon (indicating the microfilm is digitized). Here is an example of a catalog title, with the camera icon circled in red to the right of the microfilm numbers:
The last thing to do is click that camera icon to begin viewing the digitized content! As always, you can print or download any record. We're also provided with options found under Tools, to help tweak an otherwise unreadable image
Digitization efforts are still underway, so not everything is digitized and viewable yet. We should hope to see updates to these collections regularly, until they're completely digitized. I have been able to view Evangelical, Greek Catholic, Jewish, Reformed and Roman Catholic registers. Happy researching!