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The estimation of model parameters with uncertainties from observed data is a ubiquitous inverse problem in science and engineering. In this paper, we suggest an inexpensive and easy to implement parameter estimation technique that uses a heteroscedastic Bayesian Neural Network trained using anchored ensembling. The heteroscedastic aleatoric error of the network models the irreducible uncertainty due to parameter degeneracies in our inverse problem, while the epistemic uncertainty of the Bayesian model captures uncertainties which may arise from an input observation's out-of-distribution nature. We use this tool to perform real-time parameter inference in a 6 parameter G-equation model of a ducted, premixed flame from observations of acoustically excited flames. We train our networks on a library of 2.1 million simulated flame videos. Results on the test dataset of simulated flames show that the network recovers flame model parameters, with the correlation coefficient between predicted and true parameters ranging from 0.97 to 0.99, and well-calibrated uncertainty estimates. The trained neural networks are then used to infer model parameters from real videos of a premixed Bunsen flame captured using a high-speed camera in our lab. Re-simulation using inferred parameters shows excellent agreement between the real and simulated flames. Compared to Ensemble Kalman Filter-based tools that have been proposed for this problem in the combustion literature, our neural network ensemble achieves better data-efficiency and our sub-millisecond inference times represent a savings on computational costs by several orders of magnitude. This allows us to calibrate our reduced-order flame model in real-time and predict the thermoacoustic instability behaviour of the flame more accurately.