Unfortunately, as both feather and parquet are columnar-oriented files. This means that you're not able to "append" as that's only possible in row-oriented file formats. Alternatives you could look into if you want to use parquet or feather is to partition the files. For example, if you have data that doesn't change, and is generated once per day, you can write and partition based on date. It does create some overhead when reading and writing out the file, but might be a better option than re-writing the entire file each time.
As it's columnar format, you're also not able to query and only read in rows where e.g. date>2017-01-01, what parquet excels at is that you're rather able to only read in the columns you need for your analysis.