If you want to overwrite the log file each time you start your command:
parity [options] > /tmp/parity.log 2>&1
If you want to append to your log file each time you start your command:
parity [options] >> /tmp/parity.log 2>&1
And if you want to log stderr and stdout into a log file while viewing the output on your console:
parity [options] > >(tee stdout.log) 2> >(tee stderr.log >&2)
Reference How do I write stderr to a file while using “tee” with a pipe?.