Order parameter is used to describe second order phase transition. It seems that in some papers it is used in the first order phase transitions. Can first order phase transition have an order parameter? If so, how can we define the order parameter in liquid-gas transition (first order)?
Yes, there may still be some order parameters in the presence of first-order transitions. But much like free energy, the order parameter is discontinuous at the transition point. For the liquid/gas phase transitions, the relevant order parameter is the difference between the densities.
Your first answer was completely OK. Order parameters can be also used for the description of first order transitions, why not?
Think for example of water to ice transition. There is a jump when you look at the plot of density against temperature (see for example: Density of water (wikimedia)) and this is what we understand (loosely speaking of course!) under the first order phase transition (small subtlety: define the order parameter so, that it remains zero in the disordered liquid phase). This UCI lecture note about phase transitions may also be helpful.