  On an algebraic curve there are Tate symbols, which satisfy Weil reciprocity
law. The analogues in higher dimensions are the Parshin symbols, which satisfy
Kato-Parshin reciprocity laws. We give a refinement of the Parshin symbol for
surfaces, using iterated integrals in the sense of Chen. The product of the
refined symbol over the cyclic permutations of the functions recovers the
Parshin symbol. Also, we construct a logarithmic version of the Parshin symbol.
We prove reciprocity laws for both the refined symbol and a logarithm of the
Parshin symbol.
