Can parental rights be taken away if a parent is not in a child's life for several months?

It doesn't sound like you have a case yet. The phoning everyday is his attempt to maintain contact. You are actually the "bad" person her by not letting him have it. It doesn't sound like you have a custody order in place yet. Without it, he has as much right to have custody of your child that you do. I think your best bet is to file a petition for custody and child support. Somewhere down the line if you have a new person in your life who marries you, you would have a better shot at terminating his parental rights. Having said all that, when you talk with a local attorney about the custody order, ask about termination. There may be a fact that you have not mentioned that might weigh in its favor.

It sounds as if he has been trying to make contact. What you might do is petition for the missed child support. If he loses his parental rights, there will be no further child support ordered.

Under Louisiana law, parental rights can generally only be terminated by the state (for instance in a case of child abuse or neglect) or in an adoption proceeding where a parent has failed to (1) Communicate with a child, or (2) Failed to comply with a court order of support, for a period of over 180 days without just cause. Usually this ground for terminating parental rights comes about with an intrafamily adoption, where a step-parent, grandparent, or aunt/uncle is adopting the child.