You can provide padding around items inside a StackPanel by adding Margins to its child elements as shown in the sample XAML below. Margin 0,5,0,5 will put 5 pixels of padding above and below the button on the panel
<StackPanel Background="#121B25" Orientation="Horizontal" >
<Button Margin="0,5,0,5" Content="Button 1" Style="{StaticResource Button.CoolPressStyle}" Commands:Click.Command = "{Binding Start}" />
<Button Margin="10,5,0,5" Content="Button 2" Style="{StaticResource Button.CoolPressStyle}" Click="Stop" />
</StackPanel>