  Dark energy equation of state $w(z)$ parametrizations with two parameters and
given monotonicity are generically either convex or concave functions. This
makes them suitable for fitting either freezing or thawing quintessence models
but not both simultaneously. Fitting a dataset based on a freezing model with
an unsuitable (concave when increasing) $w(z)$ parametrization (like CPL) can
lead to significant misleading features like crossing of the phantom divide
line, incorrect $w(z=0)$, incorrect slope \etc that are not present in the
underlying cosmological model. To demonstrate this fact we generate scattered
cosmological data both at the level of $w(z)$ and the luminosity distance
$D_L(z)$ based on either thawing or freezing quintessence models and fit them
using parametrizations of convex and of concave type. We then compare
statistically significant features of the best fit $w(z)$ with actual features
of the underlying model. We thus verify that the use of unsuitable
parametrizations can lead to misleading conclusions. In order to avoid these
problems it is important to either use both convex and concave parametrizations
and select the one with the best $\chi^2$ or use principal component analysis
thus splitting the redshift range into independent bins. In the latter case
however, significant information about the slope of $w(z)$ at high redshifts is
lost. Finally, we propose a new family of parametrizations (nCPL) $w(z)=w_0+w_a
(\frac{z}{1+z})^n$ which generalizes the CPL and interpolates between thawing
and freezing parametrizations as the parameter $n$ increases to values larger
than 1.
