Putting parent or child company on Resume/LinkedIn?

I'm working part time job and I'm wondering if I can write down the parent company as opposed to the child company on my resume (since it looks better).

My offer letter says

(Small Company) . is pleased to offer you employment on the following terms:

Does that mean I'm required to put that on my linkedin/resume?

Or for the company, can I put Parent Company, while in job description/responsibilities I say I worked under smaller company

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