I´d like to add a few more details to Paul´s update, maybe it can help someone else:
I was not able to get POP3 working, but IMAP worked for me, this is what I did to resolve, in order:
1- disable cpanel cron job
2- check log_cron table, erase support_manager task with end_date = NULL (follow this forum topic)
3- run cron manually, in a few seconds, tickets should be created.
I added again the cronjob in cpanel every 5 minutes, but it did not work. This was the command:
/usr/bin/php /home/MY-USER/public_html/MY-FOLDER/index.php cron
As no error message was returned, I tried a few more times, but no ticket was created in support manager.
Finnaly I tried to set a cron job that called to cron URL (see docs).
4- Add cron job to call cron URL, with command:
GET -t 15m 'http://MY-BLESTA-URL/cron/?cron_key=MY-CRONKEY'
(-t refers to: <timeout> Set timeout value)
This worked for me!
Side note:
I guess my problem has to be related with how I installed mailparse extension.
As I´m using a shared hosting, I´ve installed mailparse in my user account root -> '/home/MY-USER/php/ext/mailparse', also had to copy there all php extensions (as I had to change php.ini extensions_dir value).
Maybe when running cron job command begining with '/usr/bin/php', as extensions are not there and can´t find them, mails are not parsed. (I´m guessing, not sure).
Thanks everybody for all the help.