Is there a way to use parameters in a SQL Server cursor?

What you need is 2 cursors - one for the parent and one for the child .Make sure the child cursor is DECLARED inside the LOOP not outside.it will not work if you declare outside.

eg :

DECLARE @value   VARCHAR(20);
DECLARE @someKey NUMERIC(19,0);

DECLARE main_curs 
CURSOR FOR SELECT value FROM someTable where key = @someKey;

SET @someKey = 12345;

OPEN main_curs
FETCH NEXT FROM main_curs INTO @value;
while @@FETCH_STATUS = 0
BEGIN

DECLARE CHILD_CURS CURSOR FOR SELECT VALUE2 FROM CHILDTABLE WHERE value=@value;
open child_curs 
fetch next from child_curs into @x,@y

close child_curs
deallocate child_curs

FETCH NEXT FROM main_curs INTO @value;
END

CLOSE main_curs
DEALLOCATE main_curs

Here is how you can declare a cursor with dynamic SQL, using the 'EXEC()' function. Surprisingly this does work. For example:

DECLARE @QuotedDatabase NVARCHAR(128) = QUOTENAME('ReportServer')
DECLARE @ObjectID INT = 389576426
DECLARE @ColumnName NVARCHAR(128)
DECLARE @ColumnType NVARCHAR(128)
DECLARE @DeclareColumnCursor NVARCHAR(4000)
SET @DeclareColumnCursor = '
  DECLARE ColumnCursor CURSOR READ_ONLY FORWARD_ONLY FOR
  SELECT c.Name, t.Name
  FROM ' + @QuotedDatabase + '.sys.columns c
  INNER JOIN ' + @QuotedDatabase + '.sys.types t
  ON c.user_type_id = t.user_type_id
  WHERE c.object_id = ' + CAST(@ObjectID AS NVARCHAR) + '
  ORDER BY column_id'
EXEC(@DeclareColumnCursor)
OPEN ColumnCursor
FETCH NEXT FROM ColumnCursor INTO @ColumnName, @ColumnType
PRINT @ColumnName + ',' + @ColumnType
CLOSE ColumnCursor
DEALLOCATE ColumnCursor

In another place, someone suggested using a stored procedure (compiled SQL rather than an ad-hoc script) but that doesn't work either. Here's another MWE that shows the issue fairly clearly:

/* Should print:
dbNamein=master dbNameout=master
dbNamein=model dbNameout=model
dbNamein=msdb dbNameout=msdb
*/
create procedure [TestParamsWithOpenCursorStmt]
as
begin

   declare @dbNameIn [nvarchar](255) = N'tempdb',
      @dbNameOut [nvarchar](255),
      @fs [int];
   declare dbNames cursor for
      select db.[name] from [master].[sys].[databases] db
      where db.[name] = @dbNameIn;
   while (@dbNameIn != N'msdb') begin
      if @dbNameIn = N'tempdb'
         set @dbNameIn = N'master'
      else if @dbNameIn = N'master'
         set @dbNameIn = N'model'
      else if @dbNameIn = N'model'
         set @dbNameIn = N'msdb';
      open dbNames;
      fetch next from dbNames into @dbNameOut;
      set @fs = @@fetch_status;
      if @fs != 0 continue;
      raiserror (N'dbNamein=%s dbNameout=%s', 0, 0, @dbNameIn, @dbNameOut) with nowait;
      close dbNames;
   end;
   deallocate dbNames;

end;
go

execute [TestParamsWithOpenCursorStmt];

It appears that the variable (and its value at the time) gets bound to the "declare ... cursor" rather than the open cursor.

you need to set @someKey = 12345; before Cursor Declaration such as:

SET @someKey = 12345;
DECLARE main_curs 
CURSOR FOR SELECT value FROM someTable where key = @someKey;

You seem to have the order of things wrong, and you're not actually doing anything inside the cursor?

DECLARE @value   VARCHAR(20);
DECLARE @someKey NUMERIC(19,0);

SET @someKey = 12345;   --this has to be set before its used in cursor declaration

DECLARE main_curs 
CURSOR FOR SELECT value FROM someTable where key = @someKey;
OPEN main_curs
FETCH NEXT FROM main_curs INTO @value;     -- first row is fetched

WHILE @@FETCH_STATUS = 0     -- start the loop
BEGIN

-- do something here with @value

FETCH NEXT FROM main_curs INTO @value;   --fetch the next row
END 

CLOSE main_curs
DEALLOCATE main_curs

If you want to iterate over a recursive hierarchy use CTEs, see Recursive Queries Using Common Table Expressions. You can declare your cursor over the recursive CTE, eg:

create table test (
    id int not null identity(1,1) primary key,
    parent_id int null,
    data varchar (max));
go

insert into test (parent_id, data) values 
    (null, 'root'), 
    (1, 'child 1'), 
    (1, 'child 2')  ,
    (2, 'child of child 1'),
    (4, 'child of child of child 1');
go  

declare @root int = 2;

declare crs cursor for 
    with cte as (
        select id, parent_id, data
        from test
        where id = @root
        union all
        select t.id, t.parent_id, t.data
        from test t
            join cte c on t.parent_id = c.id)
    select id, data from cte;       
open crs;

declare @id int, @data varchar(max);
fetch next from crs into @id, @data;
while @@fetch_status = 0
begin
    print @data;
    fetch next from crs into @id, @data;
end

close crs;
deallocate crs;

But most often the recursive CTEs can completely eliminate the need for a cursor.