It's difficult to find appropriate English translations for the names of various judicial bodies, especially historical ones...
To my knowledge, there has never been a parish court. Since 1971, any legal issue (whether civil or criminal) is normally first heard before a magistrates/circuit/inferior court (tingsrätt), then optionally before an appellate court (hovrätt), and finally (in cases of special interest) before the supreme court (Högsta domstolen).
Before 1971, the inferior court was the magistrates' court (rådhusrätt) in the cities and the rural district or hundred court (häradsrätt) elsewhere. Originally court rulings were appealed directly to the King, but the system of appellate courts was established in the 1600's with the creation of Svea hovrätt in 1614. Other appellate courts were created later. In 1789, the Supreme Court (Högsta domstolen) was established under the chairmanship of the King, who maintained voting rights in the Supreme Court until 1909.
To answer your question, I believe the rural district court was supposed to hear any kind of case falling within its territorial jurisdiction, the hundred (härad, often a group of adjacent parishes). The cases may have been about theft, adultery, or murder, or the court may have been asked to resolve typical village disputes between neighbours (such as who was responsible for maintaining a particular fence). In a criminal case leading to a death sentence, the execution had to be approved by the King (capital punishment was abolished in 1921 for peacetime offences and in 1972 for wartime ones; the last execution took place in 1910).
Sources: Nationalencyclopedin (NE, 1989-1996), Fyrspråkig juridisk ordbok (1995; I won't recommend it since it's mostly useless to find out anything I don't already know)