Depends what he is not doing. 6a is excellent (what year is he btw because I would expect more from a year 9 than a year 8 for example).
I would be delighted to see one of my pupils doing this well and looking to extend and improve.
I would particularly be looking for them to explain and analyse and show understanding of different interpretations of history rather than just give a descriptive narrative; and to explain why views are different on some historical events and why some people/events are seen as more important/significant than others.
I would be looking for them to use primary (from the time period) historical sources to support their work and being able to say why some sources provide better or more convincing evidence than others e.g. they are more reliable, good provenance (who said it, when etc) what the limitations are, impact of bias etc. It is very important he test the value of the source as evidence not just explain what the source tells him.
I don't think you would be seen as 'one of those parents' - I am always telling students/parents what their DC need to do to hit the next level/sub level. But do remind him what normal expected progress is, and that he mustn't be disappointed if it takes a while to get to the next level. Developing skills can take time, and History is about showing understanding, being able to analyse and explain; not about remembering lists of dates or simply describing a narrative. It is a hard subject in many respects which puts a lot of challenges in front of pupils in terms of reading comprehension, written communication and analytical abilities.
www.education.gov.uk/schools/teachingandlearning/curriculum/secondary/b00199545/history/attainment will give you the attainment targets and help you/him prepare for his next task. Remind him too that whilst it is good to aim to achieve higher grades he can challenge himself in other ways. For example if he is getting these excellent grades by always writing a report or essay, think about how he does in a speech or a presentation. Can he still achieve great grades when asked to present his work in an alternative way. Can he put his work together to make a film, complete with soundtrack and subtitles etc. He can challenge himself in many ways. It is good to try a variety of research and presentation methods as he will identify his own strengths, see what he needs to improve in and also stop himself from becoming to reliant on doing what has 'always worked' before.
Give him a pat on the back. 6a is a terrific achievement.