People are related with a type of cousin relationship if they share a common ancestor and the most recent common ancestor is two or more generations away from both people.
This means neither person is an ancestor of the other, they do not share a parent (siblings), and neither is a sibling of a common ancestor (aunts/uncles and nieces/nephews).
Cross cousin's are descendants from opposite-sex sibling.
A cross first cousin is either the maternal cousin on the father's side of the family or the paternal cousin on the mother side of the family.
At least one set of one person's great-grandparents are the great-great-grandparents of the other person.
Cousins in law are the cousins of a person's spouse or the spouse of a person's cousin. A person shares a first cousins once removed relationship with their parents' cousins and their cousins' children.
Parallel and cross cousins on the other hand are reciprocal relationships.
Parallel cousins are descended from same-sex sibling.
Cousins that are related to same sex siblings of their most recent common ancestor are parallel cousins.
A parallel first cousin is either the paternal cousin on the father's side of the family or the maternal cousin on the mother's side of the family.