We consider the problem of estimating emission line parameters from the measurements of a multi-order slitless spectrometer. This problem can be viewed as a multi-frame deblurring problem with shift variant Gaussian blur. By using Cramer-Rao lower bound theory, we derive analytical precision limits to this parameter estimation when the measurements are corrupted with Gaussian noise. The derivation involves an approximation to the Fisher information matrix in order to obtain closed-form expressions. An important feature of our treatment is to provide a framework for exploring the optimized instrument requirements, including various potential observing scenarios, spatial and spectral resolutions, and signal-to-noise ratio.