I saw a shirt for $97. I borrowed $50 from mom and $50 from dad. I bought the shirt with $3 of change. I gave $1 to mom, $1 to dad, and kept $1 for myself. Now I owe my mom $49 and my dad $49. $49+ $49=$98+my $1=$99. Where did the other $1 go?

The other one dollar didn't go anywhere!  This is a counting trick!  The $49 + $49 + $1 = $99 is a very clever counting trick to hide the remaining $1  which physically didn't go anywhere!  This is a numbers puzzle!

The error comes about by being inconsistent in how you account for where  the $3 in change went after the purchase of the shirt, i.e., first you used the $3 in change to erroneously subtract $1 from what you owe each of your parents ($50 - $1 = $49) (you're counting backwards!) and then you  added to that total debt balance of  $98 ($49 I owe mom + $49 I owe dad) the $1 which you kept for yourself to then arrive at the false sum of $99, when you should have correctly and consistently added the $3 change back onto the $97-cost of the shirt so that the correct total balance of $100 would then be realized and accounted for as follows:

Amount borrowed from mom -- $50
Amount borrowed from dad   --  $50
Total borrowed                          --  $100


Cost of shirt                              --     $ 97
plus $ 3 change:
One dollar to mom                  --     $   1
One dollar to dad                     --     $   1
One dollar for myself              --      $  1
Total                                                  $100
This problem reminds me of the counting joke/trick that someone told me years ago:

 "Do you know that you have 11 fingers instead of 10?!"  Then the person telling the joke holds up a hand and begins counting his fingers on one hand and counts backwards as follows:  "ten - nine - eight - seven - six",   and then he quickly raises his other hand and says, "and five more makes eleven!"