Your mod rewrite needs to be:
RewriteRule ^([A-Za-z0-9-]+)/?$ s.php?s=$1&p=$2 [NC,L,QSA]
instead of:
RewriteRule ^([A-Za-z0-9-]+)/?$ s.php?s=$1&p=$2 [NC,L]
The QSA at the end tells it to Query String Append and keep existing querystring values instead of stripping them off.