That’s probably outside the scope of the property management company.
I would approach it as follows: –
Send a letter to the owner, offering to split costs of the common fence. Explain that it needs to be fixed because of liability issues (neighbors kids) could open him up for a lawsuit.
If that doesn’t work – unfortunately, changing out a fence w/out sign off from the person on the other side means you need to be careful to make sure it’s 100% on your side of the property line. If you take down “his” fence, he could get icky.
(assuming he, could be a she.)
We get along with our neighbors. We’ve always split the fence bills – and we provide the labor. (Neighbors are older.) It’s an easy enough DIY job.