Assuming all files are in the same folder:

mkdir new
for i in *.html; do
    sed 's|website.com|website.com/old|g' "$i" > "new/$i"
done

If you're commonly editing website file contents you should invest in an editor.

Sublime Text 2

Drop the website onto the icon and it will build the project:

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run cmdshiftF and enter the replacement:

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Click the Replace button and verify the Replace:

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It will open and change all the files:

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BBEdit

Is an excellent and robust application I primarily use to edit code. Drag and drop the folder onto BBEdit and go to File -> Save Project and it will create a file named foobar.bbprojectd:

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run cmdshiftF and select the project with the replacement parameters:

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It will prompt you after clicking Replace All:

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If you are unsure on your changes select Leave Open because if you select Save to disk it is very hard to revert back.


Perl

There is already a sed solution but you could also do the same with Perl:

# !/bin/bash

DIR="/Users/vader/desktop/test"
NEW="$DIR/new"

if [ ! -e "$DIR/new" ]; then 
    mkdir -p new
    echo "made folder new"
else
    echo "folder exists"
fi

for htmlfile in $(ls "$DIR"/*.html); do
    TEMPFILE="$htmlfile.$$"
    htmlbase=$(basename $htmlfile)
    echo "Working on "$htmlbase
    perl -pe 's|www.website.com|www.website.com\/old|g' "$htmlfile" > $TEMPFILE && mv $TEMPFILE "$NEW/$htmlbase"
done

There are other editors out there but these are the two I primarily use. You could also do this in pure AppleScript but you haven't shown any code and mentioned terminal so I made the above in Perl.