"a part of" vs "part of"

Question

thefreedictionary.com:

“When my father is home on the weekends, the whole family always goes to a movie together.”
The dependent clause has the subject "my father", and the predicate is "is home on the weekends". The dependent clause is linked to the independent clause by the subordinating conjunction "when", which can also be considered part of the dependent clause’s predicate. The subject of the independent clause is "the whole family", and the predicate is "always goes to a movie together".

Why is it possible to write "part" without an article?
I can only understand if there were "a part of the dependent clause’s predicate" meaning "one of parts of the dependent clause’s predicate".

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Looking at similar threads on this site and on others, I could find the only one explanation on this topic. It is best expressed here:

A is part of B. – A is integral to B, A and B are indivisible
A is a part of B. – B consists of a number of components, one of which is A

According to this explanation, only my version (with "a part of") is correct:
["when", which can also be considered part of the dependent clause’s predicate] — incorrect because the dependent clause’s predicate consists of several different components ("when", "is", "home", "on the weekends") and "when" is just one of them
["when", which can also be considered a part of the dependent clause’s predicate] — correct

Answer 1

The source you quoted about the difference between the two forms is at least half right, in that "a part of" has a meaning similar to "a section of", where there's some kind of division in something that's not uniform. So if you're talking about something uniform, like a chicken breast, "a part of" is incorrect semantically:

Part of this chicken breast is undercooked.
A part of this chicken breast is undercooked.

As far as I can tell though, the reverse is not true. That's to say, I can't think of a context where it's incorrect to use "part of" to refer to a distinct portion of something non-uniform.

They only painted part of the car.
Part of the factory isn't getting power.
Part of my family lives in Europe.

This means there's apparently no rule against using "part of" when describing a part of something divisible, like a single word in a dependent clause's predicate.