How can I change the background color of my TTabSheets?

The standard Windows colour for a tab sheet is white. That standard came into being when XP themes were introduced. If a user switches back to Windows Classic then they will get a grey background. [You do mean grey rather than beige don't you? Beige would be truly vile!]

A panel inside a tab sheet can never be behind the page since it is inside the page. What is actually happening is that the panel is being drawn transparently so that the standard tab sheet colour prevails.

Use this unit in your Form at the interface:

unit MSCtrlsStyleHook;

interface

uses
  Winapi.Windows, Winapi.Messages, System.SysUtils, System.Variants, System.Classes, Vcl.Graphics,
  Vcl.Controls, Vcl.Forms, Vcl.Dialogs, Vcl.ComCtrls, Vcl.ExtCtrls, Vcl.Themes,
  Winapi.CommCtrl;

type
  TTabSheet = class(Vcl.ComCtrls.TTabSheet)
  private
    procedure WMEraseBkgnd(var Message: TWMEraseBkgnd); message WM_ERASEBKGND;
  end;

  TPageControl = class(Vcl.ComCtrls.TPageControl)
  private
    procedure TCMAdjustRect(var Msg: TMessage); message TCM_ADJUSTRECT;
  end;

implementation

{ TPageControl }

procedure TPageControl.TCMAdjustRect(var Msg: TMessage);
begin
  inherited;
  if Msg.WParam = 0 then
    InflateRect(PRect(Msg.LParam)^, 3, 3)
  else
    InflateRect(PRect(Msg.LParam)^, -3, -3);
end;

{ TTabSheet }

procedure TTabSheet.WMEraseBkgnd(var Message: TWMEraseBkgnd);
var
  LRect  : TRect;
  LCanvas: TCanvas;
begin
  if (PageControl <> nil) and StyleServices.Enabled and
     ((PageControl.Style = tsTabs) or TStyleManager.IsCustomStyleActive) then
  begin
    //Get the bounds of the Tabsheet
    GetWindowRect(Handle, LRect);
    OffsetRect(LRect, -LRect.Left, -LRect.Top);
    //create a TCanvas for erase the background, using the DC of the message
    LCanvas := TCanvas.Create;
    try
      LCanvas.Handle := Message.DC;
      LCanvas.Brush.Color:= $fafafa;// Color You need;
      LCanvas.FillRect(LRect);
    finally
      LCanvas.Handle := 0;
      LCanvas.Free;
    end;

    Message.Result := 1;
  end
  else
    inherited;
end;

end.

Set the style property to tsFlatButtons
The background ~colour~ will revert to beautiful clBtnFace