Computing parity function on n variables with O(n) gates
Because the parity is 1 if and only if the XOR of all the bits is 1.  In other words, computing the parity is equivalent to computing the XOR of the bits.  Sipser describes how to compute the XOR of the $n$ bits with $O(n)$ gates, which means that this same circuit computes the parity of those $n$ bits with $O(n)$ gates.