The short answer is no if those links are natural (not trying to manipulate SERPs).
When rel="nofollow" should be used?
rel="nofollow" should be used in such cases:
paid links, sponsored links
With rel="nofollow", you do not endorse the target website, which would be contradictory with the aim of your page. I mean, you ...
Yes, you can promote your own websites on other websites you own. Google would never penalize you for this. In fact, big sites do this all the time when they have a network of sites that go over various categories. For example, thebump.com has this promotion in their footer on all pages and at all times:
This would only become a concern if you started to ...
While Google can penalize "personal blog networks" that all link to each other, it usually has to come with spammy, low quality and/or duplicate content for Google to really care. If you're just lightly promoting one upstanding, unique website from another, there's not much to worry about.