If the mapping quality score (MAPQ) of a read is zero, then it means that the read maps to multiple positions equally well. MAPQ is a property of single reads, and is not changed by pairing. However, the mate reads can be used to attempt to chose between the alternatives, as some will be flagged "properly paired" and some not, depending on whether the position of the mate read is consistent with the way the library was constructed. In favourable scenarios, only one paired alignment is proper, and BWA will select the primary alignment accordingly. But the MAPQ of the multimapping read will stay low.
In your case, you are aligning against HLA variants, therefore there are very similar reference sequences in your index, and as a consequence you can expect a large number of alignment with a mapping quality that is null or very low. The example read that you showed is not properly paired, so you really can not extract reliable information from it. I think that you should consider discarding it. Perhaps this alignment happened because sequencing errors tipped the balance in favour of one haplotype for read1, and another haplotype for read2. And perhaps you have reference sequences so similar that distinguishing between them becomes very hard. Depending on the goals of your analysis, I recommend that you look at where are such blind spots in your index.