With what should I supplicate for my parents and family?

Supplicate whatever you wish for the good of this world and the Hereafter, for yourself, your parents, your family, and all Muslims. The best supplications are those taken from the Book of Allah and the Sunnah, such as:

"My Lord, forgive me and my parents. My Lord, have mercy upon them as they raised me when I was small." "Our Lord, grant us in this world good and in the Hereafter good and protect us from the punishment of the Fire." "O Allah, rectify for us our religion which is the عصمة (safeguard) of our affairs, and rectify for us our worldly life in which is our livelihood, and rectify for us our Hereafter to which is our return. And make life an increase for us in every good, and make death a rest for us from every evil." "O Allah, we ask You for all good, immediate and delayed, what we know of it and what we do not know. And we seek refuge in You from all evil, what we know of it and what we do not know." "O Allah, I ask You for Paradise and what brings one closer to it of word or deed, and I seek refuge in You from the Fire and what brings one closer to it of word or deed. And I ask You to make every decree You have decreed for me good."

One can refer to specialized books on adhkar (remembrances) and supplications, such as "Tuhfat al-Dhakirin" by Al-Shawkani.