It appears that the Joker either did, or didn't kill Bruce Wayne's parents. It all depends on the story line. However, if we go with the Tim Burton movie Joker was the alley killer story line, how much older (in-universe) than Bruce Wayne would the Joker be? An answer in years is accurate enough. I'm not worried about a few months.
10-14 years older
If we look at the Batman wiki for Jack Napier in Burton's universe, it mentions that:
When he was fifteen, Jack was convicted of assault with a deadly weapon.
Years later, as a young man, he and his partner in crime mugged Bruce Wayne's parents, Dr. Thomas Wayne and his wife, Martha.
Since the language used mentions "years later" after he was 15 years old, and that he was a "young man", let's assume that he was between 20 and 24 (in some places, being 18 or 19 is still considered a teenager).
Now, the wiki for Batman/Bruce Wayne said:
One night when he was ten years old, Bruce had accompanied his parents to see Footlight Frenzy at the Monarch Theatre in downtown Gotham. Following the show, his father had decided they should walk for a bit before taking a cab back to Wayne Manor. It wasn't long before his mother had noticed someone following them. Attempting to evade the followers, who turned out to be two hoodlums...
So if Bruce was 10 years old and we assume from the language given that Napier was between 20 and 24, then there is a 10-14 years difference.
We can see that his date of birth (highlighted by the bottom red arrow) is "10/04/64". The mugshot was prepared on the date highlighted by the top red arrow which reads either "1/25/87" or "1/25/89" ('89 could mark the year when the film came out):
Here is a zoom-in image of the date prepared:
This implies that Napier was 25 years old before he fell into the vat and became The Joker. So if he was 25, and we assume that he killed Bruce's parents when he was 18 instead of between 20 and 24 years old, that would give us 7 years. So in 7 years time, 10 year old Bruce turned from this...
A 17 year old Batman?
However, on-screen when Alfred presents Batman with Napier's Police Case Files, it is mentioned (near the end of the wiki page) that he is in fact 38 years old instead of 25. This would make the details of the mugshot obsolete and in terms of their ages, would make a lot more sense.
The "contemporary" age of Bruce Wayne is actually relatively easy to pinpoint. Wayne always had one age at which he stopped aging in the comic book continuity, one age that was given to him to last and be one of his defining physical characteristics. Lets start from the genesis of the character Bob Kane mentioned in the late 1980's that he envisioned Bruce Wayne as being like young Cary Grant ( via BRT). Cary Grant was 35 in 1939 when Batman/Bruce Wayne was born creatively.
This is very much in line with Batman's age established in Modern continuity - 34. Batman #599 states that Bruce Wayne's parents were murdered when Bruce was nine, nearly 25 years ago. That would make Bruce 34.
One noticeable difference between the two continuities is that Modern Age I establishes that Bruce debuted as Batman at the age of 25, Year One in particular. So in contemporary timeline, Batman is purging crime from Gotham City's streets for 9 years
IN THE MOVIES BATMAN
- At the time of the shoot Adam West was 38 and finished up at the age of 40, which would make him 4 years older than the comic book counterpart at the beginning of the series
He was 41 at the time of Batman Returns
- At the time of the Batman Forever shoot Val Kilmer was 36
BATMAN & ROBIN
- George Clooney was 36 at the time of the Batman & Robin shoot. It is interesting to note that at the time Clooney described Batman as being a "35 year old guy", whether it was a lucky guess or did DC supplied him with the info on the character is unknown.
*At the time of the shoot for Batman Begins Bale was 30, making him the youngest actor to play the character in theatrical Batman movies, and making him 4 years younger than his comic book counterpart.
THE DARK KNIGHT
He was 34 on the shoot for The Dark Knight
THE DARK KNIGHT RISES
and 37 for The Dark Knight Rises
Joker's age is never directly mentioned in the comic books. The common notion is that Joker is much older than Wayne, probably stemming from the TV show and the 1989 movie, however it seems like they're the same age. The idea of Joker being older than Batman might also come from the fact that the artwork often depicted Joker as someone looking like an aged man in late 30's or early 40's, or the idea alone that he's a gangster and a leader of mobsters in the Golden and Silver eras, something that usually seasoned, older gangsters could be.
Let's again start with the genesis of the character. Joker's appearance was based on Conrad Veidt's character from the movie "The Man Who Laughs", which is a nice way of saying that Bob Kane simply drew the character on paper without any changes. Kane was a self proclaimed copycat and the Joker was virtually the Veidt character with green hair
Conrad Veidt was 35 in "The Man Who Laughs" and since Joker was an on-paper depiction of the character from the movie, it was therefore depiction of a 35 year old man. This would also more or less match the Modern Age depiction of the character. So Batman was said to start his crime fighting career at the age of 25 and Joker fell into the chemicals shortly after Batman's first appearance when he was still a gasp inducing novelty. If he's the same age that Batman is, that would make him 25 by the time he made the jump which would match the physical depiction of the character and the story - in The Killing Joke the comedian (later in continuity named Jack) is depicted as a young man who is struggling to find a job and support his pregnant wife who's expecting their first baby. So then in contemporary timeline, 9 years later, Joker would also be 34/35 and that would match the original character.
IN THE MOVIES BATMAN
- At the time of the shoot Cesar Romero was 59 and finished up at the age of 61, which would make him 24 years older than the comic book counterpart at the beginning of the series
- Jack Nicholson was 52 at the time of the Batman shoot, continuing the TV shows depiction of Joker as being much older than Batman. He was 17 years older than his comic book counterpart
THE DARK KNIGHT
- Heath Ledger was around 27 at the time of The Dark Knight shoot. He was 8 years younger than his comic book counterpart
Okay, recently (August 5, 2016) a movie called Suicide Squad came out in theaters and Jared Leto, actor and lead singer from 30 Seconds to Mars, who played Joker in it, he did an interview and said;
"I had no idea that the Joker is 70 (seventy) years old."
So in result, Joker's real age is 70 (seventy) in 2016.