This sentence can have three VERY different meanings depending on the word you put emphasis on. "WITH my parents?" (implying you want to go without them) "With MY parents?" (implying this person would much rather go with someone else's parents) or... "With my PARENTS?" (implying that another family member can be addressed instead) I dunno, it was just something cool I noticed.
"mir" puts "die Eltern" in dative form. And following the dative form of the plural "die", changes it to "den". "Meine" follows the same way, changing to "meinen".
"Meinen Eltern" is in the dative case. As nathanjohnson83 states above, use of "mit" always triggers the dative. To show the dative you put an ending on "mein". Out of all the options for endings (mien, meine, mienen, meinem, meiner, meines), the correct ending for dative plural is "meinen". http://german.morley-computing.co.uk/mein.php