How to change the code in Arduino with another Arduino?

I'm preety new to Arduino and programing.

I have an Arduino with an uploaded code which is running standalone on battery. Now I want to change
parameters such as servo positions, delays, etc. with 2nd arduino, but not when the code in the first Arduino is running.
With 2nd arduino I want to read, change and upload new parameters from/to the 1st Arduino.
The 2nd Arduino will have to have LCD and some buttons to be able to see and to change parameters.

How do I do this?
Is this achieved with reading/writing from/to EEPROM.

Please give me some pointers how and where to begin?
Do I need some extra hardware such as AVR programmer?

I hope that I explained my desirable scenario simple enough, that you could understand it.
Sorry if there are some grammar mistakes in my english.

The usual place to store this type of parameters is indeed in EEPROM.

I'm not sure why you don't want this to be done when the 1st Arduino is running. But the 1st Arduino needs to be running. Now this is not a train smash. You can reset the 1st arduino so it will execute setup(). In setup() you can send a command to the 2nd Arduino after which the 1st Arduino will send the parameters and you can store them one by one in EEPROM. If the 2nd Arduino does not reply on the command, you can either wait or after e.g. 5 seconds finish setup() and loop will be started with the old parameters.

It can also be done from loop() in the 1st Arduino. It prevents the need for resets. You can use a button that, when pressed, will send the command to the 2nd Arduino and the second Arduino will give the parameters.

And lastly you can use the 2nd Arduino to 'create' the parameters, let it send the parrameters to the 1st Arduino which stores it and when the 1st Arduino has received all modified parameters it will start using them.

How do your Arduinos communicate?

OK. 1st Arduino needs to be running.

I think that the last option that you mentioned might be the one I am looking for. Do all my parameters that I want to change have to be in setup() or can they also be in loop()?

My Arduinos are not yet connected because I don't know which is the best or right way. Maybe through serial connection? Maybe with AVR programmer? No clue.

Please point me in the right direction.

All is depending on your concept.
Serial communication will need to modify the sketch in the first Arduino. It is a lot of programming. The communication and data processing must be included in new APP. Serial in not necesserily for such model. The I2C, ISP... can be used also but serial is relatively easy and with good range.
ISP communication, like programming the EEPROM with the second Arduino, always needs the reset = interrupt in work for the 1st Arduino. You need not extra HW. The Arduino is able to do this job.
I would go to serial, definitely.

Maybe I went ahead of myself, because I did not yet finished the coding in the 1st Arduino.
But when I have it complete, I want to proceed with communication between both Arduinos.
I will go with serial if you prefer it.
Do you know some examples or commands that I will have to use to accomplish what I want?
I want to read through the command references and try to understand what I am writing in compiler. I just dont know where to begin.

Here is: Serial - Arduino Reference. The IDE contains the examples for serial communication (also here on this web).