that palindrome is of form L=WW^R. so,here if you want to push and pop of stack first you need to find what is the center.
for example,in L=WcW^R,it is obvious that after seeing 'c',you can now pop the stack....but it is not like in the case of WW^R.so,here,actually in palindrome(even),at center the symbols should be same(e.g:aba.aba........here at center i.e.,i put the dot to represent it,....symbols are same....but it need not always the center where two consecutive symbols are same....
so,you should go with NPDA to check this,...