The target has total cover from the attack
The Core Rulebook doesn't cover this exact situation very well. As you say, you can draw lines parallel to, but not passing through, the wall. So it is possible this only provides cover, not total cover. My ruling is that the wall provides total cover in this situation and I will justify my reasons for it.
RAW Support from the Hero's Handbook
There is a diagram on page 57 of the Hero's Handbook (from the Beginner's Box) that covers this exact situation. I will try to translate their pretty diagram here:
........ * = Wall
N 2....... . = Open Space
W E *TT***3. 1,2,3 = Attacker Number
S *TT****. T = Target
Paraphrasing their annotations for each attack.
- Target does not have cover from attacker 1
- Target has cover from attacker 2 because at least one line passes through the wall.
- The target has total cover from attacker 3
Attacker 3 in this scenario is identical to your example. The book goes on to state the following about total cover:
If you are completely hidden behind a wall or other obstacle, enemies can't attack you. If every imaginary line you can draw from your corner to the enemy's corner goes through the cover, you have total cover from that enemy and you can't attack each other.
Now you could argue that the Beginner's Box is an overly simplified version of the rules and it doesn't apply to the full game. It does use very similar wording about the imaginary lines from corners however, and it is intended to serve as an introduction to the core rules. So it is certainly RAI for this situation to provide total cover even if RAW is slightly unclear.
RAW Support from D&D 3.5
3.5 edition D&D is the immediate predecessor to Pathfinder and its ruleset is mostly compatible. In the Player's Handbook for 3.5e on page 139 we have the following text (emphasis mine):
Two creatures can see each other if they can trace at least one clear straight line from any part of one creature’s space to any part of the other creature’s space. The line is clear if it doesn’t cross or even touch squares that block line of sight.
The qualification that the lines cannot touch squares that block line of sight clears this up. Also note that this does not mean that the creatures have partial cover if they both move 1 square north. The rule is any part to any part not all parts to all parts. The lines can go diagonally to avoid touching the wall.
Thanks to HeyICanChan in the comments for this one.
My interpretation of Core Rulebook RAW
With the Core Rulebook RAW situation being unclear the best I can do it provide you with my interpretation of the rules. This entire situation hinges on the single line which runs from the north edge of the attack to the north edge of the target. The argument that this does not provide total cover it because the line passes by the cover but does not "cross" it.
I would argue that if a line could be consider to not "cross" cover by only passing along the intersection between the cover square and the next, then the line also does not intersect with the targets square by the same logic. Therefore the wall should provide total cover, is the creatures are considered to be filling their entire square so should the cover.
Use DM discretion
As there is no clear determination either way you can use some situational discretion. If the target is hiding around the corner, attempting to use the wall as cover they are unlikely to leave an arm or a leg sticking out. In that situation I would rule that the wall provides total cover. Conversely, if the target is ducking in and out of cover to fire back at the attacker then you could easily rule that it only provides cover not total cover.
Use logic and common sense to determine a ruling that feels right based on the situation. You mentioned in the comments that the Hero's Handbook example depicts a jagged wall that passes beyond the grid lines. That is the sort of thing you can use to help determine the correct ruling here. Perhaps jagged cave walls provide total cover where smooth castle bricks would not? However you rule is unlikely to make a major difference as the creatures can easily move slightly further into or out of cover to correct the situation.