They're not supposed to--whether it's home warranty companies, home inspectors, title companies, insurance companies, lenders, etc.
Having said that, there sometimes is common or shared ownership. That's the case where I'm at with the title company, an insurer, and a lender. We as agents are required to disclose this relationship. And, yes, I'm sure our parent company profits if those particular providers are used. However, the agents don't get any sort of referral fee. Most agents strongly encourage their clients to check with other providers. Usually it turns out that the related providers (in the case of the company I'm with) are very price-competitive and give good service. But clients are absolutely free to use whoever they want.
Hope that helps.