I think the meaning of "peaceable" here is the same as in "peaceable possession" of real estate (by a tenant), meaning "acquiesced in by all other persons, including rival claimants, and not disturbed by any forcible attempt at ouster nor by adverse suits to recover the possession or the estate" (Black's Law Dictionary).
http://thelawdictionary.org/peaceable/

So it means that Party B's use of Party A's trademarks will not be challenged or impeded.

The following is on use of a right of way:

"It is not so straightforward, in this case, to ascertain whether use was peaceable or, by contrast, contentious. [...]
Use which continues when people have been challenged verbally or by notices is not peaceable use."
https://www.gov.uk/government/uploads/system/uploads/attachm...