Is your Time dimension connected to a source or is it a Server time dimension which has no table for it in the Data source
because usually when you generate a time dimension using BIDs it gives you these two options
and it looks like in your case you dont have a table for it in the datasource, i mean you chose a server time dimension which creates a dimension in Analysis server itself.
so you wont be able to append or add more dates, you will need to recreate it from scratch.
but if you chose the other option "Generate a time table in the data source" This option creates a time dimension table in the underlying data source and populates the same with data based on the inputs provided.
you should have access to be able to create a table in your data source
in this case you can add records to the table and reprocess it.
here is an easy script to create and fill a time dimension table
and later you can generate your time dimension in the cube from this table
and here is 2 articles about generating a time dimension and its options