Wrexham took the lead in the 19th minute when January signing Smith raced onto a Mark Howard ball over the top, before slotting home to score his first goal for his new team.
Oliver Rathbone doubled the visitors' lead three minutes later when he scored a scrappy goal as the ball eventually went in off the post.
James McClean's corner then hit the post, before Rathbone fired just wide as the Welsh side deservedly went into the break two goals to the good.
Northampton improved in the second half, which also involved a scuffle late on between both teams, but never really looked like threatening the Red Dragons' goal at Sixfields Stadium.
Parkinson's men remain third in League One, one point behind second-placed Wycombe Wanderers and eight adrift of runaway leaders Birmingham City.