Parametric decay instabilities (PDIs) lead to the generation of strong frequency-shifted radiation when powerful X-mode polarized microwaves, injected for electron cyclotron resonance heating (ECRH) of fusion plasmas, cross a region of non-monotonic electron density near the second-harmonic upper hybrid resonance (UHR). For the standard second-harmonic X-mode ECRH scenarios used at the ASDEX Upgrade tokamak, the second-harmonic UHR occurs near the plasma edge, meaning that PDIs occur in connection with phenomena leading to non-monotonic edge electron density profiles, e.g. blobs, edge-localized modes (ELMs), and inter-ELM modes. We present the first study of strong signals near half the ECRH frequency in fusion-relevant plasmas and demonstrate their PDI-like features, such as an ECRH power threshold required for their occurrence, through analog modulations of the ECRH power. The signals near half the ECRH frequency are found to be far more prevalent than expected based on previous theories, and we hence present a qualitative explanation of their origin, in terms of cross-polarized microwaves generated through non-linear interactions of the waves involved in the PDIs, which captures the basic features of the experimental observations. We additionally present examples of the PDI-related microwave signals just below the ECRH frequency which indicate the location of the microwave source. Furthermore, based on a non-linear magnetohydrodynamics simulation of an ELM crash in ASDEX Upgrade, performed using the JOREK code, the duration of the PDI-like microwave spikes observed in connection with ELMs is shown to match the transit time of an ELM filament though an ECRH beam. Finally, the PDI power threshold expected theoretically, calculated using the electron density and temperature profiles from the JOREK simulation, is shown to match the experimentally observed values.