The straightforward reading, however, is that it is forbidden to use proxy services. You're also not allowed to run them, but that's specified separately.
No that's not a straightforward reading at all.
Lets drop the 'or run' to simplify it slightly and read that:
use dedicated, stand-alone equipment or servers from the Premises that provide network content or any other services to anyone outside of your Premises local area network.
now lets apply some plain structure:
use [ [dedicated, stand-alone equipment or servers] from [the premises]] that[ [provide network content or any other services] to [anyone outside of your premises local area network]]
now lets simplify a bit more:
use [ [servers] from [the premises]] that[[provide services] to [anyone outside of your premises]]
Clearly that reads one of two ways, either you are prohibited from providing services to others from your premises, or you are prohibited from using services from your premises that are reachable from outside your premise.
The first reading makes perfect sense.
The 2nd reading prevents you from accessing anything on the internet, unless it only reachable by you, which is ridiculous.
Clearly the straightforward reading is that the prohibition is on 'using' or 'provisioning' something on-premises that provides services to others.
Using a proxy service hosted off premises is not covered by that at all.
Further, any interpretation which does read as prohibiting the use of external proxy services would also prohibit the 'use' of external email servers, and web hosts, which is plainly ridiculous.