How do we know that elementary particles possess definite parity?
From the fitting of experimental data.
Here is a review from 1965 , when we were still discovering the plethora of particles and started classifying them according to their quantum numbers.
Since spin and parity are closely related quantities, there is usually some advantage in discussing them together. The methods which have been successfully used to determine them differ widely according to the nature of the particle, the manner of its production, and its decay mode. In their simplest form the arguments involve only such general concepts as angular momentum conservation, parity conservation, and Fermi or Bose statistics. Restricted to these assumed properties, however, our knowledge of particle spins and parities, particularly for the less accessible recently discovered states, would be extremely limited. Further assumptions involving the dynamics of the transformation such as contained in continuous energy dependences, reasonable form factors, and simplest matrix elements greatly extend the analytic method at our disposal. We exhibit in this review various methods and results, referring the reader to the original papers for further discussion and qualification of original experiments.
Parity conservation was a working hypothesis until the weak interaction was found to violated it.
The electromagnetic and strong interactions are invariant under the parity transformation. It was a reasonable assumption that this was just the way nature behaved, oblivious to whether the coordinate system was right-handed or left-handed. But for several years physicists had puzzled over the decay of the neutral kaons, which had equal mass but decayed to products of opposite parity. In 1956, T. D. Lee and C. N. Yang predicted the nonconservation of parity in the weak interaction. Their prediction was quickly tested when C. S. Wu and collaborators studied the beta decay of Cobalt-60 in 1957.
But I do not see why they MUST have a definitive value of parity: they might as well be a linear combination of eigen-states.
Well, the experimental evidence says otherwise. (It is not like neutrino flavors for example).