Recent spacecraft observations reveal the continuous presence of energetic tails in solar wind and pickup ion spectra in the quiet, slow low-latitude solar wind. To explain this phenomenon, we take into account growing evidence from observations, theory, and simulations that MHD turbulence in the solar wind has a dominant two-dimensional (2-D) component. Of special significance is that 2-D MHD turbulence simulations feature turbulent field-aligned electric fields due to turbulent convection of 2-D magnetic field fluctuations. Such electric fields have been observed in the low-latitude solar wind. We explore pickup ion propagation and acceleration between the Sun and Earth with a newly developed kinetic numerical model. It includes a random distribution of large-scale field-aligned electric field fluctuations consistent with observations in the low-latitude solar wind. The main result is the qualitative reproduction of observed accelerated interstellar pickup He+ spectra at Earth. We find that a spectral knee connects the pickup ion core with the accelerated tail. Like observations, the simulated tail is approximately a power law f(v) ∝ v-4.5 until it reaches particle speeds ~10 times the solar wind speed, where a rollover occurs. The energetic tail particles originate closer to the Sun, where acceleration is most efficient. The knee forms because the local pickup ion source particles are abundant but weakly accelerated. Further predictions, such as large knees for pickup ion species formed from interstellar neutrals with large ionization cavities, and the absence of knees in accelerated spectra of inner source pickup ions are also discussed.