  Using parametric equations of state (relativistic polytropes and a simple
quark bag model) to model dense-matter phase transitions, we study global,
measurable astrophysical parameters of compact stars such as their allowed
radii and tidal deformabilities. We also investigate the influence of stiffness
of matter before the onset of the phase transitions on the parameters of the
possible exotic dense phase. The aim of our study is to compare the parameter
space of the dense matter equation of state permitting phase transitions to a
sub-space compatible with current observational constraints such as the maximum
observable mass, tidal deformabilities of neutron star mergers, radii of
configurations before the onset of the phase transition, and to give
predictions for future observations. We studied solutions of the
Tolman-Oppenheimer-Volkoff equations for a flexible set of parametric equations
of state. We compare tidal deformabilities of stars with weak and strong phase
transitions with the results of the GW170817 neutron star merger. Specifically,
we study characteristic phase transition features in the $\Lambda_1-\Lambda_2$
relation, and estimate the deviations of our results from the approximate
formul{\ae} for $\tilde{\Lambda}-R(M_1)$ and $\Lambda$-compactness proposed in
the literature. We find constraints on the hybrid equations of state to produce
stable neutron stars on the twin branch. For the exemplary equations of state
most of the high-mass twins occur for the minimum values of the density jump
$\lambda = 1.33-1.54$; corresponding values of the square of the speed of sound
are $\alpha = 0.7-0.37$. We compare results with observations of gravitational
waves and with the theoretical causal limit and find that the minimum radius of
a twin branch is 9.5 - 10.5 km. For these solutions, the phase transition
occurs below 0.56 [fm$^{-3}$].
