Ordinarily one would call a party for a first birthday a “first birthday party”, or might write “first-birthday party” if concerned about ambiguity. Phrase “first year birthday party” would not be used. Phrase “first-year birthday party” could be used, and avoids the slight ambiguity mentioned before. (A 3-year-old baby having his or her first party for a birthday could be said to be at a “first birthday party”. While few people would use the phrase that way, some people might write “first birthday-party” to intend such meaning.)
Only after completion of a year are all parties (birth, marriage, etc.) celebrated. That ensures that you have completed that number of years and never take the running number of the year into account.