This could be attributed to couple things:
Is the solder finish HASL? As solder finish may not be applied to clean copper and the HASL thickness could be plugging the via holes.
Other options which could affect solder via hole intrusion.
Solder aperatures in the stencil are allowing solder to enter the hole.
Solder is wicking from pad of logic device, into via hole. You might allow your assembler or PCB fabricator to solder mask the holes. Tenting the vias on the topside will prevent solder paste intrusion in to the pads.