Can rights be taken from a parent if they are in jail?

It would probably depend on what they are in jail for. If they did harm to the child, it may be possible. Different laws for different states.

You can make a motion with the court, but the other parent will have the right to defend it and object. If he don't care, it won't be a issue, but if he objects I doubt if they would allow it.

Not permanently. They will be obviously severely curtailed while incarcerated but when released the parent is a parent and still has at least some degree of rights. How much will depend on what the court decides, based on his/her fitness as a parent, whether (s)he presents a potential danger to the child, etc.