It's a part of your duties to proofread. It's part of your duties to proofread. 分別？

a part of 或 part of 文法上都成立，只好歸類於修辭。
這問題連英語專家也難以回答。唯一勉強能夠解釋的是:

1.) Without the article it means a portion of the whole. (part of 以全體的角度來審視此一部份)
It's part of your duties to proofread.
=> Proofread is part of your duties, besides editing, verifying facts and whatnot.

2.) With the article it means a specific sub unit with defined boundaries. (a part of 將此一部份由全體中凸顯出來)
It's a part of your duties to proofread.
=> Proofread is a part of your duties and you should put more energy into it.

但即使如此的解釋，還是難以服眾。像這種有爭議性又難以有適當解釋的英語用法，只要了解兩種都不是錯誤就可，也只好暫時擱置一邊。(英語中，類似的問題多的是，想要事事都有合理的解釋，只是徒勞，又容易心生困惑。)

另參考:
http://english.stackexchange.com/questions/13711/as-part-of-versus-as-a-part-of
http://english.stackexchange.com/questions/158232/difference-between-part-and-a-part