Can a parent with shared parental responsibilities request for more time with child?

You can ask, but if you can not show a substantial change of circumstances, then under the law you are not entitled to a modification. Of course, if the mom agrees, then a Stipulation can be submitted and most Courts would be happy that the parties had worked out an arrangement.

The questions do not revolve around your feeling like you have no rights. Stop thinking in terms of your rights. Think instead of the rights of the children and what is in their best interest. Yes, you can file a petition for modification of final judgment, requesting more time-sharing, and a reduction in child support. The children have the right to be with you as much as possible. The plan would be to negotiate a better time sharing arrangement where they are with both parents for an equal amount of time. there are plenty of attorneys who deal with this precise issue frequently.

It's not that easy to just change the parenting plan - there must either be a stipulation by both parties or a showing of a substantial change in circumstances. Based on what you said in the question it does not seem like there is a substantial change in circumstances.

"The parenting plan we have now is not working out for me" is not a reason to change the current plan the burden of proof is high.