Magnetic dipole orientation and magnetic field gradient Ok, if an iron atom (or anything with a  magnetic dipole) sits somewhere in the region between two permanent magnets, the dipole will align itself with the resultant field in that position. However, the overall atom will feel a force in the direction of increase in resulting magnetic field. My question is: do these vectors always point in the same direction? That is, will the magnetic moment vector be parallel to the force that the atom experiences?
 A: 
My question is: do these vectors always point in the same direction?

I'm assuming you mean at the low temperature limit, since of course at the high temperature limit the spin will not be aligned with the external field (it will be fluctuating and with a vanishing expectation value), while the force will still exist and will still be determined by the field gradient.
Even in the low temperature limit, the answer is: No, not always.
For example, when a highly anisotropic magnetic molecule, e.g. a so-called Single-Ion Magnet, is in presence of an external magnetic field, in general its magnetic moment is not parallel to the external field, except if said field is exactly aligned along the so-called "easy axis of magnetization" of the molecule. Instead, at low temperatures the magnetic moment will be aligned along the orientation of this easy axis of magnetization. Typically, the external magnetic field will in general only (a) determine which of the two possible directions along the preferred orientation the magnetic moment takes and then (b) act as a small perturbation on this orientation.
The limit in which the answer is yes, as far as I can tell, will be that of low temperatures and large magnetic fields (or low magnetic anisotropy). This low magnetic anisotropy can be the natural situation of any atom in vacuum and in absence of an electric fields, or most molecular radicals in organic chemistry, but is definitely not the case in many inorganic magnetic molecules based on transition metal ions or lanthanide ions.
For some perspective on Single Ion Magnets, see R. Sessoli, "Magnetic molecules back in the race", Nature, 548, 400–401 (2017). For a reference of the orders of magnitude involved in this magnetic anisotropy for Single Ion Magnets, consider this boxplot of the effective barrier Ueff for the inversion of the magnetization of about 600 samples of different chemical "families" studied in the last 20 years or so. As you can see, although magnetic memories are limited to temperatures below 80K (and typically much lower), the barriers agains thermal reversal of the magnetization can be pretty formidable, even for single molecules:

Figure 1: Effective barrier Ueff for the inversion of the magnetization vs chemical "families". Obtained from SIMDAVIS (Duan, Y., Rosaleny, L.E., Coutinho, J.T. et al. Data-driven design of molecular nanomagnets. Nat Commun 13, 7626 (2022). https://doi.org/10.1038/s41467-022-35336-9).
