Maybe this should be a comment to Henning's answer, but I think there is still a crucial point of the reasoning to be explained, namely...
For each prefix of x associate a k-tuple over {0,1} that represents the parity of each of the k letters in that prefix. Obviously, there are only $2^k$ such binary k-tuples possible, so if there are $2^k+1$ letters in x, one of the k-tuples must repeat at two different prefixes -- there's the pigeonhole principle. The points at which they repeat give Henning's two prefixes in which the parity of each letter is the same.