You can add people other than your parents, just click on Add at the top right and click on New Person. If you don't add your parents then many of the useful features of Wikitree - relationships, connections, DNA connections - won't be available to you. The profiles of living people who are not active Wikitreers are always private. They show up in searches with just a surname and a range of dates to protect their privacy but also to help avoid creation of duplicates. Here's a link to the Help page for Living People. https://www.wikitree.com/wiki/Help:Living_People
Thanks everyone, this is very helpful. I thought I had to provide my parents' email addresses. I've been able to add them anonymously so now I can start building up my tree with 20 years of genealogy research, and hopefully it will be helpful for others.