Two dimensional 3 bit parity check
I'm assuming that your code works in the following way:


The input is an $n \times m$ bit matrix.
You add a parity bit to each row, chosen so that the parity of the new row is even.
You add a parity bit to each column, chosen so that the parity of the new column is even.
You end up with an $(n+1) \times (m+1)$ bit matrix in which the parity of each row and column is even.


Suppose now that between 1 and 3 errors occurred. There must be either a row or a column in which exactly 1 error occurred (I'll let you figure out why). This row or column will have odd parity, and so you will be able to detect that an error occurred.