No, there are no fields for mother, father, or parent.
See the Sentence Template Language article in Help.
You could Share a child's Birth fact with the parents.
I have Birth fact roles of Her child and His child for the parents. I picked those role names for how they print on a Family Group Sheet in each parent's records.
This is a possible sentence for the Birth fact for the Principal:
[person:full]<%, male,|, female,|, of unknown gender,> was born< [Date]>< [PlaceDetails]>< [Place]>< [Desc]><?[Her chikd][His child]| to <[Her Child:Given]>< and [His child]>| to unknown parents>.
The Gender switch prints whether the person is male, female or of unknown gender.
The Value switch prints the Her child and His child roles if there are any. It prints of unknown parents if the fact is not shared.
I didn't experiment with how the sentence might need to be changed if there is a father with no mother as I just added the Value switch for the parents to my normal Birth sentence for this experiment.
My sentence for the Her child and His child roles are the same:
[Thisperson:hisher]<%Person% son, [person],| daughter, [person],> was born< [PlaceDetails]>< [Place]>< [Date]>.
We had a thread on the forum awhile ago that brought up RM was automatically printing the parents of a person in some circumstances. I didn't find the thread and don't remember all the details, but, if you share a child's birth with parents, you should probably check the Narrative report to see if you need to customize the sentence for that person and remove the parent part.