I don't have time to look up specific statistics right now, but will edit them in when I do.
However, the general details around this:
This will vary by team, in particular based on the financial constraints of a team. The downside of calling up a top prospect before late June is that he may be a Super 2, meaning he has one more year of salary arbitration - leading to higher total salary (by quite a bit in some cases).
It also may impact when a player becomes a free agent, as a player becomes a free agent after six year's playing time - not calendar years, but season-years. A player may earn 172 days of service time in a year, from day of game 1 to day of game 162; if he earns 171 or fewer, then he hasn't completed his full year of service, so it's standard to wait at least 2-3 weeks for that first year (and a bit more if a player had a few days of service time the year before in September).
A team like the Yankees or the Dodgers, who isn't all that worried about salaries, is more likely to call up a player before the Super 2 deadline, though they usually will at least wait the few weeks to get one more year of control.
I would expect pitchers and position players are similar, in that if they are a top prospect they'll wait either way. A pitcher is more likely to have been called up for spot starts in a prior year, so might be delayed slightly more the next year.
I didn't find raw statistics available, though I suppose someone with sufficient time could comb Baseball Reference box scores. However, Fox Sports posted an article by Baseball Prospectus' Zachary Levine that included a chart of call-up dates for #1 prospects.
While you do see a decently large number called up on opening day, you see large bumps in the third bucket (when that prospect will then have a seventh year of control) and the buckets around day 70 (around when Super 2 usually hits, between 128 and 140 days, subtracted from the 183 day season means between day 43 and day 55 - hence the bump around 60-70. 17 players were called up during that bump, which is over 25% of the total 68 #1 prospects.
Additionally, a player who is optioned to the minors for over 20 days in a season then loses those days from their service time. Levine goes over the 8 star players who were called up on opening day, and only three of them - Colby Rasmus, John Danks, and Jose Fernandez - were not sent down for a qualifying period during the first few seasons. The others were sent back down, and still gained the extra year of control.