Use a regex to extract the "video id" after watch?v=
Store the video id in a variable, let's call this variable vid
Get the embed code from a random video, remove the video id from the embed code and replace it with the vid you got.
I don't know how to deal with regex in php, but it shouldn't be too hard
Here's example code in python:
>>> ytlink = 'http://www.youtube.com/watch?v=7-dXUEbBz70'
>>> import re
>>> vid = re.findall( r'v\=([\-\w]+)', ytlink )[0]
>>> vid
'7-dXUEbBz70'
>>> print '''<object width="425" height="344"><param name="movie" value="http://www.youtube.com/v/%s&hl=en&fs=1"></param><param name="allowFullScreen" value="true"></param><param name="allowscriptaccess" value="always"></param><embed src="http://www.youtube.com/v/%s&hl=en&fs=1" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="344"></embed></object>''' % (vid,vid)
<object width="425" height="344"><param name="movie" value="http://www.youtube.com/v/7-dXUEbBz70&hl=en&fs=1"></param><param name="allowFullScreen" value="true"></param><param name="allowscriptaccess" value="always"></param><embed src="http://www.youtube.com/v/7-dXUEbBz70&hl=en&fs=1" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="425" height="344"></embed></object>
>>>
The regular expression v\=([\-\w]+) captures a (sub)string of characters and dashes that comes after v=