13. A.

Notice that all two-digit palindromic numbers $11, 22, \cdots$, 99 are multiples of 11.

Since $N$ is the sum of two-digit palindromic numbers, $N$ is also a multiple of 11.

Among the multiples of 11, the smallest three-digit number is 110, and $110=77+22+11$, so $N=110$.
Therefore, the sum of the digits of $N$ is $1+1+0=2$.
13. If a number reads the same from left to right as it does from right to left, it is called a "palindromic number." For example, 12321 is a palindromic number. If $N$ is the smallest three-digit number that satisfies the following conditions: $N$ is equal to the sum of three different two-digit palindromic numbers and $N$ itself is not a palindromic number, then the sum of all the digits of $N$ is $(\quad)$.
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6