The Bayesian parameter estimation problem using a single-bit dithered quantizer is considered. This problem arises, e.g., for channel estimation under low-precision analog-to-digital conversion (ADC) at the receiver. Based on the Bayesian Cramér-Rao lower bound (CRLB), bounds on the mean squared error are derived that hold for all dither strategies with strictly causal adaptive processing of the quantizer output sequence. In particular, any estimator using the binary quantizer output sequence is asymptotically (in the sequence length) at least 10 log₁₀ (π/2) ≈ 1.96 dB worse than the minimum mean squared error estimator using continuous observations, for any dither strategy. Moreover, dither strategies are designed that are shown by simulation to closely approach the derived lower bounds.