Spectroscopic analysis relies on identifying and understanding the spectral peaks that represent unique characteristics of an analyte. In high-speed real-time settings, current peak fitting techniques, particularly Bayesian methods involving MCMC or variational approximation, can be prohibitively expensive. We propose an unsupervised method using a convolutional neural network to estimate the number of peaks and their parameters along with posterior uncertainty, by amortizing variational inference in a classical parametric peak model. In a simulated data study, our method reliably determines the number of peaks, precisely estimates parameters similar to direct variational inference, and accurately captures uncertainty comparable to MCMC methods. Our novel, fast, and precise method for Bayesian spectral analysis opens new possibilities in real-time spectral data processing for high-speed monitoring and control.