Typically, the PM is seen as an agent of the Homeowners Association.
If the crime in question dealt with the PM performing functions as the HOA agent, then yes, the Association would likely have to pay attorney fees.
Keep in mind, this is what D&O insurance is for.
Sometimes, an Association will wait to see the outcome of the case prior to making any payment agreement.
As others have said, read your contract with the PM. Then read your D&O policy.
If you still have questions, contact an attorney to answer them for you, as you are in legal territory and getting info from the net might be fine for this but a decision should be made on actual legal advice. I suspect it would only take an hour of the attorneys time. Cost: $300 or so.
Here is some info:
Things Community Managers Should Not Do In SC