He typically uses a ringflash to flatten the tone::color ratio. He also famously used a Plaubel Makina for a long time, but has used a variety of gear, including celphones. The self-reflection onto snapshots and other functions of photography in social context is a Big Deal for him.
Here's a Juan Buhler photo of Parr being snapped by my celphone:
A difficulty (ongoing) with this "discuss a...photograph" topic is that the photos are embedded in a particular context. Parr's famous advocacy of book form makes this particularly prominent.