I don't think parse_datetime has already a tryFormats feature equivalent. For that, you could use col_types argument in read_csv to set this colum as character and do the date parsing on a second step after reading (using as.Date, lubridate::ymd or anytime::anydate).
Basically, this would mean do not use readr parsing and use your own. When you want custom thing, this is often the way to go I think.
Hope it helps