When did animals begin to "parent"?
Parental care

What you call "parenting" is called "parental care" and "parental care" is any investment of the parent to improve the fitness of their offspring. Parental care can take many forms from niche building for the offspring to parental feeding passing by placing plenty of ressources in the egg.

Issue with the definition of parental care

Given the vastness and eventual lack of a strict enough definition of the concept of parental care, it will likely be difficult to really know what you would consider as being parental care. Below I am suggesting two possible cases. If you were expecting to another answer, you will need to provide a nice and strict definition of parental care.

Mutation rate

Under a very broad definition of parental care, one could argue that first examples of parental care would be found in very early organisms (bacteria/archae, eventually viruses) that would at some point (for a given lineage) reduce its mutation rate to ensure transmitting a good genome to its offspring cells.

Nutrients in eggs

If you feel unhappy with me using such a broad definition, you might want to consider the first examples of parental care as being the existence of anisogamy (see here), where females (by definition) produce large eggs filled with nutrients, proteins, mRNA and other things. I doubt we know what lineage first evolved but it could be an ascomycota eventually.