Checking Cambridge Advanced Learner's Dictionary shows that the British and American pronunciations for each of esplanade and promenade differ. Also the way the ending is pronounced for each dialect is different in the two words. Are there any explanations for this phenomenon?
In English spelling, a silent e at the end of a word usually signifies the preceding vowel should be pronounced as a "long vowel". Some of these words borrowed from other languages are pronounced according to the rule (/eɪd/), others are pronounced more closely to the borrowed word (/ɑːd/).
Both of your words have a French origin.
There are many other such words in English with a variety of pronunciations, e.g.:
I don't think there is a rule governing the pronunciation of these words. Perhaps some speakers have been influenced by greater exposure to French. That would be consistent with my impression that US speakers prefer the 'anglicised' form. I find their anglicising approach more logical, as for instance, nobody pronounces grenade: /grəˈnɑːd/.