If someone looked at the house without an agent, the new listing agent gets the full commission (both sides) if that buyer uses the new listing agent.
The holdover period (could be 3 to 6 months) from the old agent goes away when the home is relisted with a different company/agent.

Well, here's the exact situation... My parents realtor is terrible - their contract with her is up 5/31 (today). They told her 2 weeks ago they were not renewing their contract with her. There is someone who has looked at the property with their agent and my parents agent has been in contact with their agent. She says they are interested but it's been 2 weeks and supposedly the buyer has been ill and hasn't been able to get with his realtor to write up a contract. So, if they list effective 6/1 with a new agent, and then let's say on 6/10 that buyer writes up a contract to buy the house, who gets the commission - the new agent? Or the one who had the house listed when they looked at the house a month ago?

The contract with agent #1 should detail what the terms are for the situation you describe. Dig out the contract and happy reading!

If your parents want to be really fair, they'll agree to pay Broker #1 if the house sells to Buyer Smith.

They can exclude Buyer Smith from the new listing agreement.

I wish you the best.

MLS rules dictate what happens when a listing expires and gets relisted with a different company/agent.
I don't know of any MLS that would pay the "expired" listing agent if the buyer has had their own agent.
The new listing agent and the buyers agent get paid.

"....let's say on 6/10 that buyer writes up a contract to buy the house".
You mean the buyers agent writes the contract?