Is it permissible for a person to say, "Ya Aba" (Oh Father) when tired, claiming that their intention is not to seek help, and is it appropriate to denounce this statement?

This type of invocation is metaphorical and is not intended as a call or an appeal for help, and it is not forbidden to utter it. If it is not intended as an appeal for help or assistance, then it is not a reprehensible act. However, if that is intended, then it is a grave reprehensible act; because it involves diverting worship to other than Allah. Most likely, the speaker does not intend that, and the phrase is commonly used to lament one's state and to feel compassion for oneself.