This paper focuses on channel estimation in single-user and multi-user MIMO systems with multi-antenna base stations equipped with 1-bit spatial sigma-delta analog-to-digital converters (ADCs). A careful selection of the quantization voltage level and phase shift used in the feedback loop of 1-bit sigma-delta ADCs is critical to improve its effective resolution. We first develop a quantization noise model for 1-bit spatial sigma-delta ADCs. Using the developed noise model, we then present a two-step channel estimation algorithm to estimate a multipath channel parameterized by the gains, angles of arrival (AoAs), and angles of departure (AoDs). Specifically, in the first step, the AoAs and path gains are estimated using uplink pilots, which excite all the angles uniformly. Next, in the second step, the AoDs are estimated by progressively refining uplink beams through a recursive bisection procedure. For this algorithm, we propose a technique to select the quantization voltage level and phase shift. Through numerical simulations, we demonstrate that with the proposed parametric channel estimation algorithm, MIMO systems with 1-bit spatial sigma-delta ADCs perform significantly better than those with regular 1-bit ADCs and are on par with MIMO systems with high-resolution ADCs.