Can we have a three-level system with mixed parity and transition dipole moments in different directions? I would like to know if there are molecules described by a three-level system with mixed parity and transition dipole moments in different directions. By mixed parity I mean all three transitions can be excited by electric field. In principle I don't think such a system is forbidden, but it would be perfect if there are some examples. Does anyone have an idea? Thanks! 
 A: Yes, this is perfectly possible - just choose a molecule that's asymmetric enough and you'll get mixed-parity eigenstates (because if the Born-Oppenheimer electronic hamiltonian is not parity-symmetric, there's no parity requirements on the eigenstates), which likely means that all the dipole transition matrix elements are nonzero.
If you want a specific example of a triplet of levels with nonzero transition dipoles between all pairts, try e.g. water, whose molecular orbitals look like this,

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so that for your triplet you can therefore take the $2a_1$, $3a_1$ and $4a_1$ orbitals, which will have nonzero transition dipoles on all three transitions. On the other hand, though, all three transition dipoles will point along the same axis.
If you want to break that symmetry and get transition dipoles in three different directions, then you just need to take an even more asymmetric molecule. In terms of simple examples, I would go for things like $\mathrm{C\,H\,F\,Cl\,Br}$ or $\mathrm{C\,H\,F\,Cl\,OH}$, though as soon as you stray from the very simple molecules the standard collections no longer house the molecular orbitals and you need to break out the GAUSSIAN (or equivalent) to nail down the orbitals and the dipole transition matrix elements. 
As one likely candidate I would suggest  $\mathrm{C\,H_2\,F\,OH}$, which is in the standard list in molcalc.org, and which has no set symmetry,

so any pair of molecular orbitals will yield a nonzero transition dipole pointing in some non-symmetry-dictated direction.

Now, as to whether this will give you a useful set of states, that's a separate matter. If you want to restrict your attention to three states in particular and ignore the others, you typically need a good reason for it. In atomic physics this is normally justified because you can tune your lasers very precisely to those transition energies, so you know that you don't have population 'leaking off' to the rest of the available state space. (This is what was referred to in the comments as 'suitable for use as a qutrit'.) 
For a molecule this is still possible but it becomes harder, because the lines are made broader by the coupling to the nuclear degrees of freedom, which often feel under no symmetry obligations to follow any selection rules, and which have very tightly spaced manifolds of non-equispaced nuclear eigenstates.
This comes with the territory when you're doing molecules, though, so it's all a matter of finding a threesome of levels that are addressable enough, and less an in-principle deal of finding molecules which are asymmetric enough for the electronic transition dipoles to be released from their symmetry constraints.
