What is a possessory conservator?
The possessory conservator has the same rights and responsibilities as any other parent except s/he does not decide the primary (main) home of the child. The judge can limit these rights at his/her discretion.1 One parent is usually made possessory conservator when the other parent is made the sole managing conservator.2
However, a judge will not make a parent possessory conservator if it is not in the best interest of the child and might harm the physical and emotional well-being of the child.2
To read about the rights and responsibilities of a possessory conservator see What is conservatorship (custody)? What types are there? and What is possession of and access to a child (visitation)?
1 Tex. Fam. Code § 153.192
2 Tex. Fam. Code § 153.191