Does the the context, parallel texts, earliest translations, or earliest commentaries explain, affirm, or refute whether Exodus 34:27 actually indicates the existence of an Oral Law?
Note: For grammar and syntax aspects, please see related question - which might inform the answer to this one.
Jewish Encyclopedia, Oral Law - According to the rabbinical interpretation of Ex. xxxiv. 27, the words indicate that besides the written law——God gave orally to Moses other laws and maxims, as well as verbal explanations of the written law, enjoining him not to record these teachings, but to deliver them to the people by word of mouth (Giṭ. 60b; Yer. Meg. iv. 74a; comp. also IV Ezra [II Esdras] xiv.).
NASB, Exodus 34:27 - Then the Lord said to Moses, “Write down these words, for in accordance with these words I have made a covenant with you and with Israel.”
Jewish Virtual Library, Judaism: The Oral Law -Talmud & Mishna - Without an oral tradition, some of the Torah's laws would be incomprehensible.