All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
First of all, we thank and praise Allaah Who guided you to Islam and we ask Him to make you firm and steadfast on His Religion until the Day you meet Him. We also ask Allaah to enable your parents to embrace Islam.
You should continue to call them to Islam and ask Allaah to guide them to it; this is one of the best acts of kindness and dutifulness from you to them.
As you mentioned in the question, the fact that one of your parents or both of them are non-Muslims is not an obstacle for being kind and dutiful to them, and it is permissible for you to visit them. However, it is forbidden to sit on a table where intoxicants are being served. In that situation, you are not permitted to sit with them. You may apologize to them in a nice and soft manner in order to avoid any bad reaction from them.
If we presume that they get angry because of you refusing to sit with them on a table where intoxicants are served, then you should be patient and try to solve the matter with wisdom. The best example to follow are the companions of the Prophet who were faced with the same situations but this did not lead them to be kind to other people at the expense of their religion.
Moreover, sticking to the principle may be more beneficial and may bring better results, as this reflects to the other party one’s sincere adherence to his/her religion. Indeed, this may well be of greater effect on the person and a reason for his embracing Islam.
As regards the necessity which renders what is forbidden permissible, it is when one is predominantly sure that he will die or face an unbearable difficulty; in this case what is forbidden for him becomes permissible.
The best example for this is what happened to ‘Ammaar ibn Yaasir when he was severely tortured and he uttered the word of disbelief [in order to prevent further torture] while his heart was content with faith. Allaah accepted this as a sound excuse; undoubtedly that was a case of necessity.
As regards some inconveniences or disturbances which may happen to a person, then one should be patient, as this earns him great rewards. However, soon these disturbances would vanish; the daily life proves this and what you already experienced from your parents before is an evidence proving this.