Not just parish records.
New Hampshire state-level vital records exist on FamilySearch, Ancestry (two databases, one of which appears to be the FHL film, and the other a full colour reproduction), AmericanAncestors. In addition, I started research in the pre-internet days, and actually did research in the Archives physically searching the card files, and have references to the drawer number.
With other American vital record data sets, it seems that Ancestry is chopping them up into city or county segments. While that makes browsing easier (especially in looking for someone who doesn't show in the index because of mistranscription or a misspelling in the original record), it doesn't give a good way to cite the record.
The problem with citing a link is that it requires the reader to have access to whatever web site is linked, and assumes those links will be permanent. I have had numerous occasions of trying to verify trees posted by someone else, only to find that the links they cite are dead links.