  In \cite {FrKn,Sbornik} it was shown that in some knot theories the crucial
role is played by {\em parity}, i.e.\ a function on crossings valued in
$\{0,1\}$ and behaving nicely with respect to Reidemeister moves. Any parity
allows one to construct functorial mappings from knots to knots, to refine many
invariants and to prove minimality theorems for knots. In the present paper, we
generalise the notion of parity and construct parities with coefficients from
an abelian group rather than $\mathbb{Z}_2$ and investigate them for different
knot theories. For some knot theories we show that there is the universal
parity, i.e.\ such a parity that any other parity factors through it. We
realise that in the case of flat knots all parities originate from homology
groups of underlying surfaces and, at the same time, allow one to "localise"
the global homological information about the ambient space at crossings.
  We prove that there is only one non-trivial parity for free knots, the
Gaussian parity. At the end of the paper we analyse the behaviour of some
invariants constructed for some modifications of parities.
