If the date is unknown, send DbNull.Value as the parameter's value:

If dateIsUnknown Then
    cmd.Parameters.Add(New SqlParameter _
                       With {.ParameterName = "@purDT", _
                             .SqlDbType = SqlDbType.Date, _
                             .Value = DBNull.Value})
Else
    cmd.Parameters.Add(New SqlParameter _
                       With {.ParameterName = "@purDT", _
                             .SqlDbType = SqlDbType.Date, _
                             .Value = theDateVariable})
End If

Or if you prefer,

cmd.Parameters.Add(New SqlParameter With {
                   .ParameterName = "@purDT",
                   .SqlDbType = SqlDbType.Date,
                   .Value = If(dateIsUnknown, DBNull.Value, DirectCast(theDateVariable, Object))}
                  )

It is necessary to cast the variable to be of type Object so that the If operator has a common type to return. The Value parameter expects an Object.

It depends on the data method you are using to send the data to the server.

purDate is a variable of DateTime type and it cannot be set to null.

I suggest you use IsDate instead of testing length.

Dim purDT As Date
    If Not IsDate(TextBox1.Text) Then
        purDT = Nothing
    Else
        purDT = Convert.ToDateTime(TextBox1.Text)
    End If

You could use the Nullable(Of Date) to allow your purDT variable to also be Nothing :

Dim purDT As Nullable(Of Date) = Nothing

If tb2.Text <> "" Then
    purDT = Convert.ToDateTime(tb2.Text)
End If

However, the magic happens when you define the SQL parameter that will hold this value. The parameter should either be DBNull.Value or a valid (non-null) Date :

' initialize connection, command...

Dim param = New SqlParameter()
param.ParameterName = "NullableDate"
param.Value = IIf(purDT Is Nothing, DBNull.Value, purDT)

cmd.Parameters.Add(param)
cmd.ExecuteNonQuery()

Try this :

Dim purDT As Nullable(Of Date)

If tb2.Text = "" Then
    purDT = DBNull.Value
Else
    purDT = Convert.ToDateTime(tb2.Text)
End If