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Equivalent Potential Temperature, defined as

$$\theta_e = \theta \cdot e^{qL/c_pT}$$

with $\theta$ denoting potential temperature and q the specific moisture of water remains constant for wet adiabatic processes. Does it make sense to assign it also to unsaturated air parcels? Lets say, an an unsaturated parcel is lifted to LCL, where condensation starts. Is $\theta_e$ constant during this dry adiabatic process too? I'm not sure, but I would say, that T must be replaced by the T', at which condensation occur - otherwise it makes no sense.

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Yes. Since the later term doesn't change during dry adiabatic processes and the former neither, the equivalent potential temperature $\theta_e$ does also stay constant for dry adiabatic processes. Whether it makes sense to "assign" (maybe you meant "measure") it to unsaturated air parcels depends on the application.

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