Difference between 部门 and 系
系 means  college major and is exclusively used by the education system. 专业 is a synonym in this context.

For a student of the English major, he would say


  我读英语系。


or


  我读英语专业。


An institution, a faculty or a big department that hosts multiple majors is usually translated into 学院. In a college, there can be only one language department which teaches multiple languages as different majors.

So when you say 


  我在英语系工作。


It means you work for the English department (if there is), or you work on stuffs related to English major for the language department. While


  我在语言学院工作。


Means you work for the department/school of language.

部门, or 部 for short, is usually connected with the roles from an (commercial) enterprise, such as HR Department (人力资源部门), Operations Department (运营部门), Sales Department (销售部门), IT Department (信息技术部门), etc. 

Sometimes you can connect 部门 with the 'business subject'. It's not the best word choice but is acceptable, such as 洗衣机部门, 冰箱部门, if you are a manufacturer and have dedicated sector for each product.

Back to the education system, you should stick with 系, 专业 or 学院 mentioned above. '英语部门' sounds very weird to me. However you can say '语言学院的IT部门' meaning the 'IT office for the language department'.

As a recap, 系 and 部门 means different things and are not interchangeable in this context.

Update:

When I wrote this answer, I didn't realize there is a difference between Chinese college organizational model and the U.S. one. 

To clarify, the education system in China used to follow the Russian model, that is, there are different offices for different majors in one department, and there are supporting staffs like administrators and operation personnel within the department, each of whom may be assigned to serve a specific major or all majors in the department.

Now they are gradually switching to the Western model, that is, each major is a department, and supporting services are run by the college.