What is the best way to stop weeds from growing in a mulched area?

Remove the mulch and save. Then lay down a sheet of black plastic or several layers of old newspapers. If using old newspapers I would suggest to spray them with a mixture of vinegar and salt before replacing the mulch. After laying the plastic sheeting or newspapers replace the mulch.

Or get some rocks to place on the area.

I use Preen every three months

You could also place some pots of flowers on the rocks if you want some color around the bushes.

I know plastic seems to be a good answer, but please learn from the previous homeowner and my extensive labor. The roots of your shrubbery won’t get water. The plastic stops it. So in our case, the azaleas sent roots above the plastic and being about 40 or more, it took years to get it all out. Weeds still grew. Unless you have something like wiregrass or bermuda invading your beds, the weed seeds are on top of your mulch. Rain or watering gives them a chance to be covered and they take root. To use the Preen, which is a pre-emergent, you will have to get rid of the weeds first.

Salt and vinegar, while both good weed killers, can kill your shrubs as their roots may take it in. Our neighbor uses a propane tank and attachment to burn the weeds in his drive and in beds, but not too close to the shrubs or buildings.