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The statement "My mother is my parent" is always true, however, the opposite statement "My parent is my mother" is not always true because my father is also my parent.
What's the right term in logic for this phenomenon? And what are the fallacies that overlook this phenomenon called?
The difference is that between inclusion and equality.
The set of Mothers of x is included into the set of Parents of x, but not vice versa:
every Mother is a Parent, but not every Parent is a Mother.
In terms of Aristotelian logic, we have that "being a Parent" is predicated of every Mother.
We may also say that the concept Mother is subordinated to the concept Parent.
One could say that being your mother is sufficient, but not necessary for being your parent.
Thus, thus phenomenon could be called "sufficiency" (without necessity).
Fallacies that overlook this are affirming the consequent, in which one infers being your mother based on being your parent, and denying the antecedent, in which one infers not being your parent based on not being your mother.
In linguistics, the words mother and father are hyponyms to the word parent and parent is a hypernym to mother and to father.
In logic, the proposition x is the mother/father of y entails that x is a parent of y but parenthood does not entail motherhood/fatherhood.
The simple answer is that motherhood implies parenthood, while parenthood does not imply motherhood.
Or: being a mother implies being a parent, while being a parent does not imply being a mother.
So "implication" is the right term that applies to this relation.
A quick observation : in the OP's example, the statement "my parent is my mother" is true but not complete.
A false statement would be "my ONLY parent is my mother" - but that was not stated nor does the actually stated form imply that "my parent is my father" cannot also be true as one person (conventionally) has either two parents or is a clone -case in which the team of scientists and the "parent" organisms can be seen as "mother(s)" (?).
Anyway, the pedantic thing to call that could be "assimetry by omission" in which one would adress elements of the subclass "mother/father" in relation to the descriptor in the superclass "parents" that includes it. Generalization by omission, or simply stating parts of complex truths without implying completitude (not a lie by omission), are weaker forms. In extremis there is nothing wrong with the example given from a strictly logical point of view " my mother is my parent" =true, but the other part of the truth " people (usually) have two parents" = also true does not change the first conclusion.
