What should the converter parameter be for this binding

I would name the control and then bind using ElementName:

<UserControl x:Class="BaySizeControl.BaySizeTextBox"
    xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
    xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"  
    xmlns:local="clr-namespace:BaySizeControl"
    Name="Foobar"
    >
    <UserControl.Resources>
        <local:BayListtoStringConverter x:Key="BaySizeConverter"/>
    </UserControl.Resources>
    <Grid>

        <TextBox  Name="Textbox_baysizes" 
                  Text="{Binding RelativeSource={RelativeSource self},
                                Path=Parent.Parent.BaySizeItemsSource, 
                                Converter={StaticResource BaySizeConverter,
                                ConverterParameter={Binding ElementName=Foobar} }}"
                  />
    </Grid>
</UserControl>

No, wait, that won't work because the ConverterParameter is not a Dependency Property, nor is the Binding a DependencyObject. A ReleativeSource markup extension should do what you want, though I've not used it nested inside other MarkupExtension - perhaps it is not well behaved in this case:

<TextBox  Name="Textbox_baysizes" 
                      Text="{Binding RelativeSource={RelativeSource self},
                                    Path=Parent.Parent.BaySizeItemsSource, 
                                    Converter={StaticResource BaySizeConverter,
                                    ConverterParameter={RelativeSource self} }}"
                      />