Car Parinello advantages and disadvantages
Car Parinello is an Molecular Dynamics (MD) model. The MD models of chemicals I've worked with use a projection of the expected wave-function using classical mechanics to approximate what the atoms and molecules in the model are doing. Essentially assigning a force constant based on distance between the atoms in the system to approximate the energies involved in the simulation. This kind of simulation can give semi-qualitative numbers to energies of association or dynamics of a reaction in some cases where the motions of all the atoms are essentially classical in nature. However, if there is proton motion involved in the reaction coordinate or if you would like chemically meaningful energies of association MD models will not be workable for your goals. MD models can, typically, only get a "feel" for whether or not an association is possible or if a reaction could take place. And if there is a quantum mechanically allowed and classically forbidden step, say proton tunneling, the model will fail.

So MD models are great for very large or distributed systems. For example modelling a solute in solvent or large solid systems, both of which are beyond most ab initio Quantum Mechanics (QM) calculations due to the number of individual calculation steps needed for convergence. There are some "high level" (nit-picky) calculations that require precision beyond what a classically built model can give which is why DFT and ab initio calculations are still run today.

Now I have not ever used Car Parinello in my work, (I'm part of one of those nit-picky groups) so I cannot give any real numbers for the errors in this method as compared to other MD simulations. I would, however, say that the writer of the Wiki page is not un-biased (neither am I, see above regarding nit-picky group) and is trying to show how great this method is compared to other MD simulations.  This particular MD simulation seems to be using some SCF calculations and long range basis set functions for outer shell electrons and so I think it is mainly for use in solid ionic systems rather than the small isolated molecules my group typically models. 

For DFT (or essentially all modern QM calculations) the electronic wave function is approximated using basis functions. These are pre-programmed functions that can be mixed together with little computational effort to approximate the true wave function. The basis sets used by Car Parinello are plane wave basis function which means the wave function repeats at long range and allows (forces) assignment of electrons far removed from one another. This means that a self consistent state is quickly reached for solid crystalline systems. 

My personal bias (use a grain of salt for this section) is against such basis sets due to the fact that it forces a pseudo-crystalline pattern on the system (at least at long range). For most chemically interesting systems it is the oddities on a surface or the internal structure which give rise to chemical change. And so, this model may fail to represent these systems well due to the choice of basis set. Once again, not having used this model I cannot say it would fail to represent these interesting features, I just doubt it would give reasonable energies or wave functions after completing the calculations.