Brilliant question, Olga!

Answer: No article.

Explanation: It's a phrasal verb, like "to take part (in)." The word "part" is being used as an adverbial particle, not as a noun. You can compare it to German phrasal verbs using "teil," where the lower-case "t" is indicative of an adverbial particle, as opposed to the noun "Teil." You can also compare to the romance languages, which have similar expressions, also with no article. Eg: Spanish "ser parte de" if it were a noun, it would take the indefinite article, as in English.

Note: It is also possible, although much less common, to use "part" as a noun after "to be," in which case you would use the article. Of course, you're free to use any noun after the verb "to be."