Depending on the equipment/pedals in each signal chain it could result in severe phase issues. For example if you use the Origin "Bassrig" pedals which blend absolutely HORRIBLY with a parallel clean DI.
With DI + mic'ed amp (as in REAL amp) you can at least move the mic if needed.
So no, I would never run two parallell DI's unless it was a thoroughly tested setup, and even then I would only consider doing it with a FOH engineer I know and trust 100%.
For recording... still no. I often use amp+DI, but recording two parallel DI boxes doesn't make sense to me, for the reasons explained above. If I wanted something like that I would rather do the parallel processing "in the box" with plugins, where I would have more control over potential phase issues. It doesn't take much phase shift to COMPLETELY change how the bass "sits" in the mix.