Distinguishing between my cousin's kids and my second cousin's parents

No, unfortunately it doesn't exist because of the way it's derived.

First cousins share a grandparent, second cousins share a great-grandparent, third cousins share a great-great-grandparent, and so on. The "removed" indicates the number of generations separating the relatives.

So your second cousin's kids and parents are both your second cousins once removed. Your cousin's kids (I'm assuming first cousins) are your first cousins once removed. Their parents are too, but they are better known as aunts and uncles.

When you're describing first cousins, you can make a distinction based on whether your parent's siblings were the same or opposite sex. A parallel cousin is either your mother's sister's child or your father's brother's child (same gender siblings). A cross cousin is either your mother's brother's child or your father's sister's child (opposite gender siblings).

I think a simpler way would to add a qualifyer of who is of the more senior generaton.

Why? Well: a first cousin once added is the {child of your cousin} or the {parent of your second cousin}, but consider that whatever you are to them, they are the other!!

So to say "they are my first cousin once added" suffixed with "i am the senior" would clarify that you mean your cousin's child/children; "I am the junior" would imply that they are your parents first cousin.