When I was 15 (just turned 15 a week or so before), my family went away and I babysat my 3 silbings every night (it was a business trip for my dad) - they were 10, 8 and 6 and one night, I also watched 2 of my coworkers kids - they were in the 5 and 7 range. My parents didn't blink twice - I routinely watched them at home and it wasn't a big deal. I was a little older (2 years than your DC)- so we were allowed "out and about" the resort - BUT this was before cell phones - so we didn't have that added "comfort" for my parents.
There are some 13 y/o who couldn't handle it - but many could - you know your child best. If they can handle making sure the door latch is on and remembering the key in the case that they HAD to leave the room - they'll be fine.
Also - PP - fire alarms go off at home - strangers show up at the door - there are just as many, if not more, things that can go wrong at home.
If the 13 y/o is mature and is PREPARED for the fact it is different than home and how to handle the different situations, they'll be just fine. A 13 y/o watching one's own siblings is different than watching strangers- they are used to be alone together.