Pronunciation of "esplanade" and "promenade"

Solution 1:

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.

promenade (1560s) leisurely walk; (1640s) place for walking; (1887) dance given by a school
esplanade (1590s) large level area

There are many other such words in English with a variety of pronunciations, e.g.:

comrade (1590s) one who shares the same room
facade (1650s) front of a building
charade (1776) long talk, chatter

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/.