ps -eo ppid= | grep -Fwc $pid
If your grep does not support -w:
ps -eo ppid= | tr -d '[:blank:]' | grep -Fxc $pid
or ps -eo ppid= | awk '$1==ppid {++i} END {print i+0}' ppid=$pid
ps -eo ppid= | awk '$1==ppid {++i} END {print i+0}' ppid=$pid
or (clobbering the positional parameters)
set $(ps -eo ppid=); echo $#
Note that this is not atomic, so the count may be wrong if some processes die and others get spawned in the short span of time it takes to gather the data.