Live so that you wouldn't be ashamed to sell the family parrot to the town gossip.

Rogers’ quip is a homespun restatement of the idea that character shows in what you can safely have repeated about you. A parrot mindlessly echoes what it hears at home; the “town gossip” amplifies and circulates it. To “live so” that you could sell such a parrot without shame is to behave in private as you would wish to be known in public—avoiding hypocrisy, cruelty, or petty dishonesty that might be exposed by loose talk. The humor depends on a rural, small-town social world where reputation travels fast, but the moral is broadly applicable: integrity is what survives when your words and actions are repeated by others.