Behold the following:
Nothing in this agreement gives either party a license or other right to use the trademarks …
If the idea is that neither party is allowed to use the trademarks in the specified manner, say so:
Each party shall not use the trademarks …
For one, it’s shorter. And second, if you say “Nothing in this agreement gives” whichever party whatever right, that leaves open the possibility that that right might arise through some other mechanism.