I run my photography website with a VPS, but I'm looking to expand my offerings through a third-party photo hosting service Pixieset. Pixieset allows users to connect their Pixieset account/galleries to a custom domain so that clients don't have to visit two different websites/feel like they're "leaving" the photographers website.

For the sake of example, my website URL is ooiimages.com. Let's say I create a subdomain galleries.ooiimages.com connected with Pixieset.

My question is, will linking visitors to this external host as a subdomain *from* my own website via menu link result in SEO penalties for linking externally? HostGator says "Your subdomains will be treated as entirely separate websites in the eyes of Google.?|" (source: https://www.hostgator.com/blog/what-are-subdomains-affect-seo/)