We study photon-ALP conversion by resonance effects in the magnetized plasma of galaxy clusters and compare the predicted distortion of the cosmic microwave background spectrum in the direction of such objects to measurements of the thermal Sunyaev-Zeldovich effect. Using galaxy cluster models based on current knowledge, we obtain upper limits on the photon-ALP coupling constant g of less than or similar to O (10(-11) GeV-1). The constraints apply to the mass range of 2.10(-14) eV less than or similar to m(ALP) less than or similar to 3.10(12) eV in which resonant photon-ALP conversions can occur. These limits are slightly stronger than current limits, and furthermore provide an independent constraint. We find that a next generation PRISM-like experiment would allow limits down to g approximate to O (10(-14) GeV-1), two orders of magnitude stronger than the currently strongest limits in this mass range.