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We have implemented and evaluated a positron emission tomography (PET) demonstrator using two monolithic detector blocks operating in coincidence with dedicated application-specific integrated circuit (ASIC) readout. Each detector is composed of a monolithic lutetium yttrium orthosilicate (LYSO) scintillator coupled to a pair of Hamamatsu S8550-02 APD arrays. The front-end electronics of this demonstrator is based on the VATA240 ASIC readout, which sums the charge provided by each row and column of the APD array. The ASIC has been characterized obtaining a noise per row or column less than 2000 electrons rms with the APD at its inputs and a good linear response in the range from 5 fC to 30 fC. We have acquired energy spectra of 22Na and 137Cs radioactive sources, achieving energy resolutions between 13.2% and 14.1% full width at half maximum (FWHM) at 511 keV. We have estimated the interaction position over the surface of the monolithic blocks using Neural Networks (NN) position determining algorithms, obtaining spatial resolutions at the detector level down to 2.1 mm FWHM. By using this detector technology and electronics we have achieved images of point sources with spatial resolutions as good as 2.1 mm FWHM for filtered back projection (FBP) reconstructions methods with single slice rebinning (SSRB). Based on the results obtained with this demonstrator, we are developing a PET insert for existing magnetic resonance imaging (MRI) equipment, to be installed in a collaborating hospital and used for clinical PET-MRI of the human brain.