Molecular hydrogen (H2) is the simplest neutral molecule and its ionization and dissociation energies are key benchmarks for testing first-principles calculations of nonadiabatic, relativistic, quantum-electrodynamic and finite-nuclear-size effects in molecules. To obtain improved H2 ionization and dissociation energies, we performed precision measurements of 15 transition frequencies in para-H2. They connect the X 1+g (v = 0, N = 0) ground state, the long-lived EF 1+g (v = 0, N = 0) excited state, and several high members of the np and nd Rydberg series with principal quantum number n in the range between 48 and 70. The new data by itself leads to a five times more accurate value of the ionization energy of the ground state of para-H2, equivalent to a relative frequency accuracy of 5 x 10^{-11}. Combined with a global network analysis of 45 energy intervals in para- and ortho-H2 that have been measured in the past three decades, we obtain new statistically robust values for the ionization energy [EI(0, 0)/(hc) = 124 417.491 1573(58) cm^{-1}] and the dissociation energy [D0^{N=0} = 36 118.069 6646(58) cm^{-1}] of H2. These values have an unprecedented accuracy equivalent to 175 kHz. The network analysis also leads to improved term values and dissociation energies of the excited states involved in the analysis. By comparing the results with first-principles calculations of the level structures of H2 and H2+, a significantly reduced upper bound of 540 kHz is derived for putative beyond-the-standard-model contributions to the energy difference between para- and ortho-H2.