Read data from an array in another class

Try this:

public class Parsedata
{
     private string[,] m_ParticipantX;
     public void ParsedataMethod()
     {
       ...
       m_ParticipantX = new string[40, 4];//Neccesary data are added to this array
     }

     public string[,] ParticipantX
     {
          get { return m_ParticipantX; }
     }
}

using Crestron.ActiveCNX;
public partial class Form1 : Form
{
    ActiveCNX cnx;
    cnx = new ActiveCNX();
    Parsedata data = new Parsedata();

    private void SerialSend_Click(object sender, EventArgs e)
    {
        data.ParseDataMethod();
        int number = 8;
        for (int i = 1; i < number; i++)
            cnx.SendSerial(i, data.ParticipantX[i, 1]); //
    }
}

First, keep in mind that property are part of class, and class has property (or field). You cannot get a property without the class owner.

In static, you can achieve as such:

public static class Parsedata{
  public static string[,] StringX{get;set;}
}

However when your class and property is not static, you need to instantiate the class to an object first. Example:

public class Parsedata{
  public string[,] StringX{get;set;}
}
public class Caller{
  Parsedata p = new Parsedata();
  public void SetStringX(string[,] stringX){
    p.StringX = stringX;
  }
  public string[,] GetStringX(){
    return p.StringX;
  }
}

However you must keep in mind that instantiated object's value are different between instances. Example (just illustration):

WebSite a = new WebSite();
a.Name = "Stack";
WebSite b = new WebSite();
b.Name = "Overflow";
Console.WriteLine(a.Name); // will result Stack
Console.WriteLine(b.Name); // will result Overflow

The ParticipantX array is declared inside a method, which means that it will be local to that method.

You must place the declaration outside the method:

public class Parsedata
{
    public string[,] ParticipantX;

    public void ParsedataMethod()
    {
        ...
        ParticipantX = new string[40, 4];

I've marked ParticipantX as public, so that you can access it from your form:

ParseData parseData = new ParseData();
parseData.ParticipantX[x, y] ...

A better method is to make it a public property:

private string[,] _participantX;
public string [,] ParticipantX
{
    get { return _participantX; }
}

You could pass it in via a delegate + event handler, for example:

public partial class Form1 : Form
{
    public string[,] ParticipantX;
    public Form1()
    {
        InitializeComponent();
        Class1.SendArray += new EventHandler<MyArgs>(GetArray);
    }

    public void GetArray(object sender, MyArgs ea)
    {
        this.Invoke(new MethodInvoker(
            delegate()
            {
              ParticipantX = ea.myArray;
            }));
    }

    private void Form1_Load(object sender, EventArgs e)
    {
        Class1 t = new Class1();
        t.ParsedataMethod();
    }
}

public class MyArgs : EventArgs
{
    public string[,] myArray { get; set; }
}

And inside of your parse data class, call OnArraySend when you want to pass the array to your form:

public static event EventHandler<MyArgs> SendArray;
    public void ParsedataMethod()
    {
        string[,] ParticipantX = new string[40, 4];
        OnArraySend(new MyArgs() { myArray = ParticipantX });
    }

    protected virtual void OnArraySend(MyArgs ea)
    {
        if (SendArray != null)
        {
            SendArray(this, ea);
        }
    }

This is a crude example but you get the general idea

You can either make Parsedata and ParsedataMethod static methods or create an instance of Parsedata in your Form:

var pd = new ParseData();

Then you can use:

pd.Participant(...);