Suppose Mr. Parker is a brigadier general in the army. could we simply refer to him as "General Parker". In other words, can "General" be used as a generic title for anyone with a high rank in the army?
General can be used as a generic title for anyone in the Army of rank O-7 or above. In the cases of 1-, 2-, and 3-star generals, their formal titles are Brigadier, Major, and Lieutenant General, but just plain General can be used, particularly verbally. "We need to clean up around here; General Parker is paying us a visit this afternoon" is a legitimate way to refer to the 1-star. In writing, shorthand is often used to prevent ambiguity, so the same remark in an email might read:
We need to clean up around here; BGen Parker is paying us a visit this afternoon.
When saluting BGen Parker in the parking lot, "Good morning, General" would be an appropriate verbal greeting – assuming the local time hasn't passed 1200 hrs.
From a Marine Corps customs and courtesies guide (p. 9):
In written correspondence, both formal and social, full rank precedes the name and is written out. In conversation, all generals are General...
You didn't say which army, and in the British Army things are slightly different from the rules J.R. gives for the US Army. The NATO equivalent to US Army O-7 is OF-6, one-star general; the British Army equivalent to this rank is Brigadier, and
While the corresponding rank of brigadier general in many other nations is a general officer rank, the British Army considers it a field officer rank.
Perhaps as a consequence, British Army brigadiers are addressed not as 'General', but as 'Brigadier':
How to Address a Brigadier
The recommended social style of address is as follows:
Verbal communication: Brigadier Jones*
*A younger man, or a more junior officer in any of the Armed Forces, addresses him as 'Sir'.
You have to be a Major General or above to get addressed as General (Debrett's again)
So the answer to your question depends on which army Mr Parker is in.
In the U.S. Army (50 years ago, anyway, when I was a service member), a general officer was so called because he could be assigned to command Army forces of any type; hence, generals' uniforms did not indicate a corps affiliation (Engineers, Signal, etc.). "Generalist" was the idea.