Whoever is the hostess of any party, including this one, may choose whether to have a formal party or an informal party. She would use formal names on a formal invitation, and informal names on an informal invitation. An engagement party — and for that matter, a wedding — can quite properly be either one.
Now, when you say both parents are hosting, what do you mean? Are they paying for the room and the catering? Paying is not the same as hosting. Hosting means making the decisions and taking the final responsibility for the safety, comfort and entertainment of the guests. The “final responsibility” properly rests with just a single hostess — the buck cannot stop in two different places. A nice way to resolve this, is for one lady to be the hostess along with her husband (if she has one) and for the other lady to be one of her guests of honour. If it were your mother acting as hostess, a formal invitation would read
Mr and Mrs Father Bride<br />request the pleasure of the company of<br />Mr and Mrs Guest<br />to meet <br />Mr and Mrs Father Groom<br />Mr Danseur Groom<br />Ballerina <br />on Saturday the thirty-second of June &tc
the equivalent proper informal invitation would read:
Dear Jane and John,<br />I am having a little affair on June thirty-second to celebrate Ballerina’s engagement, and to introduce her fiance, Danseur Groom, and his parents, Lydia and Foxtrot Groom, to our circle of friends. Tom and I do hope you can join us, at the Centre Street Legion, at eight o’clock.<br />yours sincerely,<br />Mary
Equally informally, if your mother has a visiting card with her name engraved on it, or engraved informal notes with her name on them, then she just writes above her name
To meet Mr and Mrs Father Groom<br />Mr Danseur Groom<br />Ballerina
above her name on the card (or on the inside of the informal note) and the details of the party under her name (or on the inside of the informal note).
If you have actually managed to find a way to make one buck stop in two places, and the mothers actually are going to co-host somehow equally, then you would adapt the proper formal wording by moving the other parents’ names up to the top of the invitation, so:
Mr and Mrs Father Bride<br />Mr and Mrs Father Groom<br />request the pleasure of the company of<br />Mr and Mrs Guest &tc
You would adapt the informal invitation by listing all four names instead of the initial “I”, and mentioning “we” instead of “Tom and I”, so your mother would send to her circle of friends:
Dear Jane and John,<br />Mr and Mrs Father Groom, and Tom and I are having a little affair on June thirty-second to celebrate the engagement of our children Ballerina Bride and Danseur Groom, and introduce our family and friend to one anothers. We do hope you can join us, at the Centre Street Legion, at eight o’clock.<br />yours sincerely,<br />Mary
Danseur’s mother would write similar notes to her circle of friends, swapping around the names so that your parents get their full formal names, and she refers to herself and her husband by their informal names.