Do all children have an equal opportunity to achieve in school when the socioeconomic indicator is parental occupation?
In general, the answer to this is yes. Most children whose parents are in the same occupation or occupational category will have roughly similar life chances. This is not completely true, of course, but it is generally true.
People who are in the same occupation tend to be broadly similar to one another. They tend to have the same general level of educational achievement. They tend to make similar amounts of money. Therefore, they tend to be culturally similar to one another. When this is the case, their children will tend to have broadly similar opportunities. Their schools will generally be similar to one another in terms of the resources that are available to them. Their home life will be roughly similar, providing the same level of enrichment opportunities (or lack thereof) that tend to help children do well academically. For these reasons, children whose parents are in similar occupations will generally have similar opportunities to achieve in school.