Starring: Ron Howard, Don Knotts, Andy Griffith, Frances Bavier
Continuity mistake: When Opie and Arnold rush into the house and tell Andy and Aunt Bee about the party, in the shots facing the boys we can see the reflection in the mirror behind them, and there's a lamp atop the credenza, but in the shots facing Andy and Aunt Bee that lamp doesn't exist.
Opie's Group - S8-E9
Andy: Clara, sometimes a parent can't see what he should do, and sometimes it takes a person from the outside to show him. And I'd like to thank you.
Clara: Groovy.
Trivia: When Gomer and Barney are waiting at the Miracle Salve Co. Office, Barney speaks in a nasally high pitch voice. When Gomer asks Barney what's the matter with his voice, Barney explains that he's been talking like that as a disguise, to which Gomer asks, "How do you do that? I wish I could do my voice like that!" Jim Nabors does precisely that for his role as Gomer.
Question: Why is it when they were on the show the Darling sons never spoke? Would they be paid more if they did, or was this some sort of joke?
Answer: It was a joke. They were supposed to be dumb hillbillies.
Both. According to later interviews with both Andy and the Dillards band (the real name of the band that played the parts of the Darling children, of which Charlene (Maggie Peterson - Mancuso was also a member) the reason the Darling boys never had any lines was because they would've had to have been paid if they spoke. In return for appearing without lines or pay, Andy promised to get as much of their music onto the show as possible which, he did. This obviously gave them a lot of exposure.