MP3 player, that remembers current possition in all playlists or folders
Yes, there is.
MortPlayer Audio Books and Smart AudioBook Player have both been verified by me to have asked feature implemented.

Some details:


MortPlayer Audio Books is completely free (it displays ads, but these can be... disabled in settings), but offers an enormous number of options, settings and preferences, which can overwhelm many of fresh users; it is also based on skins, with many skins having... well... not to good UI, look &amp; feel,
Smart AudioBook Player is much simplier and offers more clean, standard UI, but it is not free. It comes with 30-days trial offering all options, after which it must be purchased for $2 or will block most extra options and features.


Both players have the most desired option. You can jump between folders or around media library and every time you open up a folder (book), that was opened by you before, you will always start listening from the position, you left it, not from the beginning.

There are probably many more audio players for Android, that supports these feature. Only, to find them,  one must search for audiobook player, not for MP3 player, as I did previously.