/etc/crontab or any user crontab is not being executed
Solution 1:
Check you don't have a /etc/cron.allow file or /etc/cron.deny file causing jobs to be ignored.
If /etc/cron.allow is not there then everyone can run cron jobs. If it is there but empty I am not sure if that allows anyone to run cron jobs.
Solution 2:
The is an error in your crontab, look at:
* * * * * root run-parts /bin/date >> /data/date.txt
You invoke run-parts with the directory /bin/date - surely this will fail.
The error goes to stderr and isn't cached by >>