Replace a specific part of a string
If the strings are all the same, and the column is called c1 in a table called t1 then SUBSTRING_INDEX() might get you there.  Don't update your table until you've verified it with SELECT. 

Whitespace added for clarity.

SELECT CONCAT(
  SUBSTRING_INDEX(c1,'&amp;',1),
  '"',
  SUBSTRING_INDEX(c1,'"',-3)
) FROM t1;


SELECT the substring left of the first '&amp;' counting from the left, a literal " and then the substring right of the third " counting from the right.

The only caveat is that the link format you've shown seems wrong.  The first "&amp;" seems like it should have been "?" instead, and you should be clipping it at the "?" which marks the start of the query string in a URL.  Perhaps youtube was doing something nonstandard with URLs to work around plugin or browser issues.