What you will normally see with a "p=" will be:
which goes to:
However, if you have:
To have it go exactly where it went before you would put:
Now, where it went before may or may not be where the original author meant it to go. With the "#p11444" it positions to the specific post, and without it you see the entire topic from the top. That is the same on the old and the new forum. So if all you are doing is adjusting links to point to the new forum, you don't add the "#pNNN".
Seeing a "p=NNN" without a "#NNN" is suggestive that the original author of the link made a mistake, but that is impossible to know without more information. They might have left the "#NNN" off on purpose, or by accident.
On a related note, if you see "&sid=HHH", where HHH is a fairly long hexadecimal number, you can remove it (or leave it alone). The "sid" portion is ignored on both the old and new forums.
One more complication, just to be through, if you see something for the form:
with viewforum instead of viewtopic and an "f=NN" portion, you need to fix those up completely by hand. The NN numbers following "f=" have changed. You need to figure out which forum/topic area they were trying to link to and find out the correct URL. Fortunately, links of this form are extremely rare.