The snippy waitress at the drag show is played by Courteney Cox's then-brother-in-law Alexis Arquette, who had previously played a Central Perk customer in The One with Rachel's Sister (2000).
Chandler's father is the last parent of the main characters to be introduced (discounting Phoebe's adoptive mother, who died by suicide before the series began).
In a previous episode, The One with Frank Jr. (1996), Chandler named a list of women he could sleep with without his girlfriend getting angry. One of them was Jessica Rabbit. Jessica Rabbit was voiced by Kathleen Turner, who in this episode plays Chandler's father.
Chandler's father's stage name is Helena Handbasket. This is the same name used by Helen in an episode of Wings when she tried to win money at an amateur drag show while visiting New York.
When Monica yells for their drinks at the drag show, the extra behind her in the blue is clearly laughing at the line said by Courtney Cox.