Naive definition of parahoric subgroup

I'm not sure what it means to define parahoric subgroups "purely in terms of $B(G, F)$"; I would say that every definition boils down to taking integral points of integral models in one way or another. By the way, it is not true in general that parahoric subgroups are full facet stabilisers; in general, the group scheme $\mathcal G$ underlying the full stabiliser is disconnected, and one must pass to its identity component before taking integral points in order to get the parahoric. (See nonetheless, say, Tits §3.5.3, or Proposition 4.6.32 of BT2, where it is observed that one does have equality for simply connected groups.)

Nonetheless, the result you want (that parahoric subgroups are pullbacks of parabolic subgroups of parahoric subgroups) is correct; it is §3.5.4 of Tits's Corvallis article "Reductive groups over local fields" (MSN), and Théorème 4.6.33 of BT2 (MSN). Of course, it doesn't really 'reduce' the problem, since one still needs to have the original parahoric subgroup to pull back its parabolic subgroups. You may also find it helpful to read Yu's various expository articles (say, "Bruhat–Tits theory and buildings" (MSN) in the Ottawa proceedings, or his paper "Smooth models associated to concave functions in Bruhat–Tits theory" (MSN; I have linked to the preprint at NUS, which I have not compared to the published version)) for a modern perspective on BT theory; he had a program for a while to make their work more accessible. He used to have some notes available on his Purdue web page, but that no longer exists, and he doesn't seem to have migrated them to CUHK.

For another perspective on parahoric subgroups, I find the appendix by Haines-Rapoport (here) quite useful. They show that the parahoric subgroup attached to a facet $\mathcal{F}$ is simply the intersection of the pointwise stabilizer of $\mathcal{F}$ in $G(F)$ with the kernel of the Kottwitz homomorphism (you may have to take Frobenius fixed points somewhere). In particular, if your group $G$ is semisimple and simply connected, then the Kottwitz homomorphism is trivial, and parahoric subgroups are just pointwise stabilizers of facets.