This is number 31,700,001. This is the approach we have taken with one special child who has a range of learning and behavioral issues, including some pretty extreme ADHD. There's no data to show it's generalizable, but we'd follow this approach were we to start over again.
The underlying assumption is that, for the truly challenged reader , this is not a race. It is the Long March. Years and years and years of work to try to reach functional literacy. Months of slow progress, inexplicable leaps forward, intermittent regressions. The primary challenge is to make the Long March something a child will manage.
1. The main direct contribution of parents is to keep the home reading experience enjoyable and regular. This means whatever one is doing at home, it brings success. School, when he's at his best and he's with the experts he wants to please, is where new learning occurs. Home is where reading is modeled and encouraged. (This advices is inverted for learners who are cooperative at home and rebellious at school. We are fortunate to have a learner who is cooperative at school and "expressive" at home.) Reading rewards have varied over the past 5 years, from stickers to chocolate to snacks to, most recently, computer time.That's about it. In the past we've struggled to bias the schools towards an evidence-based approach to reading (structured phonics is where the evidence is best) with mixed success. I think they'll get there one day, but behavioral change is very hard anywhere. In our son's case reading is so difficult I'm not sure now that any one approach would be uniquely better. Maybe his teacher's motley mix of techniques has been right for him.
2. The main indirect contribution of parents is to get the school to focus on reading to the exception of just about everything else except social skills and physical activity. Schools have a lot of mandates to meet, but our son's school has managed to focus most of his day on reading. He does some math too, which is "nice", but, really, we don't care about math, social studies, science, etc etc.
 This is in contrast to another child we have with a different set of diagnoses. He did very well with a painful Kindergarten "speech and language" program, struggled until the middle of 2nd grade, then started reading adult books. A different set of problems and a much smaller challenge.