Ultra-light dark matter may consist of axion-like particles with masses below 10 ^ { -19 } eV .
Two-photon interactions of these particles affect the polarization of radiation propagating through the dark matter .
Coherent oscillations of the Bose condensate of the particles induce periodic changes in the plane of polarisation of emission passing through the condensate .
We estimate this effect and analyze MOJAVE VLBA polarization observations of bright downstream features in the parsec-scale jets of active galaxies .
Through the non-observation of periodic polarization changes , we are able to constrain the photon coupling of the ultra-light dark-matter axion-like particles at the level of \lesssim 10 ^ { -12 } GeV ^ { -1 } for masses between \sim 5 \times 10 ^ { -23 } eV and \sim 1.2 \times 10 ^ { -21 } eV .