1. What does it mean?
부계 shortens 부계정. It is an account kept separately from a main one, perhaps for a hobby, a game, or a different audience.
Also written or said as 부계정, bugye
2. The nuance
A secondary account is not necessarily private, anonymous, or deceptive. 본계 describes which account is primary, while 비계 describes its visibility settings.
3. Examples
“그림 연습은 부계에 모아서 올리고 있어.”
I collect my drawing practice posts on a secondary account.
“이건 게임용 부계라 일상 사진은 없어.”
This is my alternate gaming account, so there aren't everyday photos here.
Sources
References support the meaning and usage. Example sentences are written for NoCapWiki.