Do I have the same rights as a married parent?
Yes. The exact same laws that govern married parents govern unmarried parents as well. Determining parenting plans, child support, custody/visitation, and the rest apply whether the parents are married or not.
However, FATHERS DO NOT HAVE LEGAL RIGHTS until paternity is proven.
When deciding on child custody and support, courts consider the same factors in cases where the parents are divorcing, dissolving a domestic partnership, or otherwise separating.
In the end, the best interests of the child ALWAYS take precedence over all other concerns. Always.