  We consider fractal curves in two-dimensional $Z_N$ spin lattice models.
These are N states spin models that undergo a continuous
ferromagnetic-paramagnetic phase transition described by the ZN parafermionic
field theory. The main motivation here is to investigate the correspondence
between Schramm-Loewner evolutions (SLE) and conformal field theories with
extended conformal algebras (ECFT). By using Monte-Carlo simulation, we compute
the fractal dimension of different spin interfaces for the N=3 and N=4 spin
models that correspond respectively to the 3 states Potts model and to the
Ashkin-Teller model at the Fateev-Zamolodchikov point. These numerical
measures, that improve and complete the ones presented in the previous works,
are shown to be consistent with SLE/ECFT predictions. We consider then the
crossing probability of spin clusters in a rectangular domain. Using a multiple
SLE approach, we provide crossing probability formulas for ZN parafarmionic
theories. The parafermionic conformal blocks that enter the crossing
probability formula are computed by solving a Knhiznik-Zamolodchikov system of
rank 3. In the 3 states Potts model case, where the parafermionic blocks
coincide with the Virasoro ones, we rederive the crossing formula found by
S.M.Flores et al., that is in good agreement with our measures. For N>=4 where
the crossing probability satisfies a third order differential equation instead
of a second order one, our formulas are new. The theoretical predictions are
compared to Monte-Carlo measures taken at N=4 and a fair agreement is found.
