Shouldn't it be "a part of work" with indefinite article "a"?
Native speaker’s answer
Rebecca
The difference between "part of" and "a part of" is subtle. In general "part of" means something that belongs to or is essential to something else. In this case, it means that going to the gym is essential to her work. While "a part of" means a piece, or a segment of something. In this case, both these terms could be used interchangeably and not affect the meaning of the sentence.