Whether you make up a new "right" or not, you can't pay someone for paternity leave if the money is already spent on government mandated BS.
You seem to be confused about the definition of the word "right" is, though. When you talk about giving people new "rights", you've made the word virtually meaningless, turning it into a synonym for "entitlement" or "privilege". This is important, very important
The right to free speech means I can voice my opinion EVEN IF THE MAJORITY OR GOVERNMENT DOESN'TLIKE IT. You have the right to a fair trial even if the government would rather ramrod you. The Bill of Rights is a list of freedoms the government "shall not infringe". Note it doesn't say "should give you", it says "shall not infringe". Rights are not at the government's discretion. They can not take your rights away because your rights don't come from the government. Your right to think your own thoughts is intrinsic to your humaness; government can only infringe your rights or not. They can not take them because they did not give them.
Your natural rights as a person include your right to have your own thoughts, and your right to your own production - to eat what you grow in your own garden, to live in the house you built.
If you build your house for your family, I do not have the "right" to kick you out and take it for my family. I do not have any right to take your food you grew in your garden. My wife is pregnant at the moment. That has zero bearing on your rights to your own food. You may choose to share some tasty vegetables with me, but knocking up my wife doesn't give me the right to take your stuff.
You want to create a new government mandate? Fine. Do not lie and call it a "right", though, because as soon as you pretend that government creates rights, government is justified in taking away your rights that "they gave you".