The answer here is
Thomas Edison really likes to keep in mind,
I missed when the name changed; Bell was the telephone, Edison was the light bulb and the phonograph.
A number that should be quite easy for you to find.
This may be a reference to finding a record, perhaps in a database. Various bits of information about you are numbers and could be stored in logs or databases; in particular, your phone number is an index into the telco's database that indicates how to route a call specifically to you, and numbers you call are similarly recorded in logs and databases for billing purposes.
Just find the sum of all the numbers you've ever called.
The hint tells us "The word sum has a double meaning here." It's not talking about adding the numbers together, it's talking about the cumulative total of all the numbers you've ever called - a record of every phone call you've ever made.
But be wary as some people might become quite appalled,
At the variant that is shared by governments abroad.
If any government, foreign or otherwise, were keeping a record of every phone call you make, a lot of people would rightly be appalled.
But remember to answer the question of whether the number is even or odd
Per OP, the fact that the solution (a RECORD) is not a number means the answer to the question of whether it is even or odd is "neither".
Think about the different types of things that Edison invented.
As noted, among other things Edison invented the phonograph.
The word sum has a double meaning here.
What do governments care about with regards to the phone numbers you have called?
They might care about you calling suspicious people or suspicious places, or using suspicious words while on the phone. See, for example, ECHELON, a communications interception system widely rumored to keep records and recordings of such things.
It is an object that can be used with an invention of Edison's. It is also something governments really like to look at carefully...
Of course, a RECORD goes with Edison's PHONOGRAPH, and governments are probably interested in the RECORDS of suspicious persons. There's also the idea of governments looking at the RECORD of calls you make, or even of actually RECORDING your calls outright.