Why did Paul and Barnabas part company before Paul's second journey?

Answer: They disagreed about John Mark

Barnabas wanted to take John Mark, but Paul did not, since Mark had deserted them in Pamphylia. Barnabas took Mark to Cyprus.

“Some time later Paul said to Barnabas, “Let us go back and visit the brothers in every town where we proclaimed the word of the Lord, to see how they are doing.””— Acts 15:36 (BSB)