Do parenteral drugs take a first pass through the liver?

No . By parenteral way may it be intravenous , intrarterial, intraosseus,intracerebral, intracerebroventricular or cutaneous the drug directly enters the blood circulation .

While the oral method makes the drug digested in the stomach and then enters the hepatic portal system.It is carried through the portal vein into the liver before it reaches the rest of the body. The liver metabolizes many drugs, sometimes to such an extent that only a small amount of active drug emerges from the liver to the rest of the circulatory system. This first pass through the liver thus greatly reduces the bioavailability of the drug.
Alternative routes of administration like suppository, intravenous, intramuscular, inhalational aerosol and sublingual avoid the first-pass effect because they allow drugs to be absorbed directly into the systemic circulaion.