Is eating of the ritually slaughtered animal permissible or not, after the father slaughtered some birds in the bathroom, then slaughtered them outside of it, in response to someone who asked him not to mention God in the bathroom?

You should keep the mention of Allah free from places of impurity and filth, and you were right in forbidding your father from mentioning Allah's name in the toilet.

As for eating what your father slaughtered: - If Allah's name was mentioned over it at the time of slaughter, then it is permissible to eat it without dispute, even if the mention occurred in the toilet. - If he forgot to mention Allah's name over it, then it is permissible to eat it according to the majority of scholars. - If he intentionally abandoned mentioning the name over the slaughtered animal, then the view of the majority of scholars is that it is not permissible to eat from it, based on the words of the Almighty: ﴿And do not eat of that upon which the name of Allah has not been mentioned, for indeed, it is grave disobedience.﴾ [Al-An'am: 121].