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Starting from a continuum-model description, we develop a microscopic weak-coupling theory for superconductivity in twisted double-bilayer graphene. We study both electron-phonon and entirely electronic pairing mechanisms. In each case, the leading superconducting instability transforms under the trivial representation, $A$, of the point group $C_3$ of the system, while the subleading pairing phases belong to the $E$ channel. We explicitly compute the momentum dependence of the associated order parameters and find that the leading state has no nodal points for electron-phonon pairing but exhibits six sign changes on the Fermi surface if the Coulomb interaction dominates. On top of these system-specific considerations, we also present general results relevant to other correlated graphene-based moiré superlattice systems. We show that, irrespective of microscopic details, triplet pairing will be stabilized if the collective electronic fluctuations breaking the enhanced $\text{SU}(2)_+ \times \text{SU}(2)_-$ spin symmetry of these systems are odd under time reversal, even when the main $\text{SU}(2)_+ \times \text{SU}(2)_-$-symmetric part of the pairing glue is provided by phonons. Furthermore, we discuss the disorder sensitivity of the candidate pairing states and demonstrate that the triplet phase is protected against disorder on the moiré scale.