Can all errors be detected? What are some examples?

No, errors can still occur. For Example (assume we are using ODD parity):
 

 Original Sequence

 Number '1' bits

 Parity bit added

        Sequence Sent

 Sequence Received

   0  1  0  1  0  1  0

     3 (even)

            '0'

     0  1  0  1  0  1  0  0

     0  0  0  0  0  1  0  0

In this example, the message sent is NOT the message received, but it would not be detected since the total number of '1' bits received is odd.