Unfortunately, it is not uncommon for someone who is asked to pay child support to lash out against the other parent who is asking for child support. And sometimes, the parent being asked for child support not only seeks some or increased parenting time, but "full custody," so that they don't have to pay child support and can use their ploy to threaten, harass, and intimidate the other parent. Judges usually see through those kinds of actions. But it's important to have adequate advice from an attorney and someone who can help you through the process even if it's unlikely the court will entertain the request. And even though he is only now requesting parenting time -- years after your child's birth and after years of not being involved -- he (and your child) are entitled to some appropriate time with each other. Know that simply because a parent hasn't previous been very involved with the child doesn't mean there shouldn't be any contact; but it certainly should -- and is -- taken into consideration by a judge when deciding what would be appropriate parenting time in your child's particular situation.
This response does not constitute legal advice and does not create an attorney-client relationship. I am licensed to practice only in Kansas. Seek legal advice from an attorney in your state or the state in which your legal claim exists.