Photoacoustic tomography is a promising imaging modality offering high ultrasonic resolution with intrinsic optical contrast. However, quantification in photoacoustic imaging is challenging. We present an algorithm for quantitative photoacoustic estimation of optical absorption and diffusion coefficients based on minimizing an error function between measured photoacoustic channel data and a calculated forward model with a multiple-illumination pattern. Unlike many other algorithms, the proposed method does not require the erroneous assumption of ideal tomographic reconstruction of initial pressures and to our knowledge is the first demonstration of the efficacy of multiple-illumination photoacoustic tomography requiring only transducer channel data. Simulations show promise for numerically robust optical property estimation as illustrated by well-conditioned Hessian singular values in 2D examples.