the drawing shows that the tub drain arm is parallel to the washer standpipe.
It could mean that it IS parallel, going vertical? This means your tub is raised up onto a split level, which maybe doesn't make sense.
...or
It could mean that it ISN'T parallel, but your drawing shows it so, just to fit a 3-d world onto a 2-d plane. In which case I think the 36" rule of thumb applies to connecting to a stack, not to another fixture's drain arm. So I think you need a vent (but it is less critical than in the case of a big drain going straight down without venting), because it is wise to put a vent between all fixtures. Years ago, some applications of Code didn't insist on venting between fixtures; this meant that proper drainage relied on "self-venting" meaning internal to the pipe, and dependent on many factors not all of which can be guaranteed, so this way of thinking is now deemed to be unreliable plumbing and not up to Code in most places anymore.
More than that I cannot say, so I hope someone else steps in here to help you out. Even how I described the situation (right above) may not be totally accurate, so someone may have a clearer explanantion. I think it is a real smart move to add in that extra vent with a SanTee, as you described.
david