The basic meanings are the same. "neither parent is coming" and "neither of my parents is/are coming" both mean "both of my parents are not coming." "Neither of my parents" is the normal, standard pattern. "neither parent" might be used in in formal writing, or if you have already mentioned both parents.
"My mom is sick, and my dad has to work, so neither (of my) parent(s) is coming."
"Children are expected to be accompanied by at least one parent. If neither (of their) parent(s) comes, the child cannot participate in the activity."
In short: it is ALWAYS OK to say "neither of my parents," while "neither parent" is only OK in a few restricted circumstances. Therefore, I recommend always using "neither of my parents."
The same is true of "either parent" and "either of their parents."