You need to separate SEO from branding for a minute.
You are saying subdomain and then presenting the www subdomain and I want to be clear that is what your intent was? If instead you are saying we have domain old-company.com and now we are going to be under domain new-company.com and we want the authority built on old-company.com then, YES, you need to do the 301 redirect of urls on old-company.com to new-company.com urls.
Now, in terms of the parent owning brand Old-Company you need to first be clear with them they do not own domain old-company.com in any way as that would negate your ability to keep the redirects once they start using it.
Google is looking at the brand only in terms of branded searches, etc. There is no branding issue beyond that. So, if the parent has domain oldcompany.com vs old-company.com or they add in The-old-company, etc. they are ok if they are ok from a business sense with that change. they do not need to have a subdomain to the previous domain like best.old-company.com
I cannot see them using the old domain in any way as being good for you. But, any variant of that old domain (not your sub domain variant with something.old-company.com) would be fine.
If I were parent and brand were a real issue, I would not relinquish the domain for any reason. If I were you and you believe you must have the redirects, I would negotiate with, "we cannot do the deal if we cannot "retire" the domain and use it as we wish except not visible on the Internet." I think you understand.
Hope that helps,
Robert