Understanding of the spatial distribution of ultrashort pulse temporal signal-to-noise ratio (SNR) characteristics at the focal spot in large aperture off-axis parabolic mirror focusing optical system is helpful for more accurate understanding of ultrafast laser pulse temporal and spatial characteristics in the ultrafast laser physics experiments. With wave-front error of optical system under consideration, the two-step focusing fast Fourier transform (FFT) algorithm with the coordinate transform based on Fresnel approximation in space domain and Fourier integral transform method in time domain are used to simulate the focusing process spatially and temporally. SNR degradation by wave-front error in large aperture optical system is explained, SNR ratio characteristics influenced by the parameters of the random wave-front error such as error magnitude and space scale are calculated and analyzed, and control requirements for different spectra of optical wave-front error of optical system are put forward. Explicating the relationship between SNR and random wave-front error in large diameter optical system is theoretical basis of designing large aperture ultrashort pulse system and wave-front error control.