Is parallel adder doing additions in each bit position simultaneously?
Each bit is added simultaneously.  However, the bit sum will not be accurate until the carry in has propagated from it's lower order neighbor.  Each carry will propagate in turn.  Therefore, you must wait until the last carry has arrived at the MSB and been included in the sum of that MSB (S4 in your example) before you consider the result S1..S4 to be stable.  At this time, you can latch or otherwise evaluate the sum.