So, let me back up just a sec so I can visualize what you are doing - you are operating in ArcMap, and you have 2 tables in ArcMap and you have selected records in one of them to 'paste' into the other?
This is probably not the only way, but since you cannot execute a 'copy' as with a feature class, pasting them into a target fc, you have the option to open your system toolboxes (there is a ArcCatalog shortcut or use the Toolbox window) and use the Append tool (since your target is an existing table).
However before you do this, let's see if Copy Rows behaves as I expect, copying only the selected rows to a new gdb table. If it doesn't copy only the selection then execute a Make Table View before Copy Rows to determine if only the selection is honored.
If this works properly, then do the same for Append processing....in other words, preprocess with Make Table View 1st if necessary. I only suggested experimenting with Copy Rows because it's easier to delete an extra table than it is to find and remove extra rows in an existing table.