today = DateList[Today];
res = {};
i = 0;
While[Length[res] < 15,
 date = DatePlus[today, i];
 ds = DateString[date, {"Year", "Month", "Day"}];
 If[PalindromeQ[ds],
  AppendTo[res, date]
  ];
 i++;
 ]
Column[DateString[#, {"Year", "-", "Month", "-", "Day"}] & /@ res]