RNA can store information just like DNA can. And it can act as an enzyme, catalyzing (for example) the production of more RNA (see here, for example).
This has lead some to postulate an "RNA world" of organisms which didn't use RNA. This is somewhat supported by the fact that a lot of the most basic nuts and bolts of the process of making proteins consist of RNA enzymes such as tRNA and rRNA.
However, if you really want to know about what you call "simultaneous evolution" you ought to look at something that would show up in the fossil record; that way you'd be exposed to more certain facts and fewer mere plausible conjectures.
Dean Kenyon went on to reject his own hypothesis, and become a prominent YEC and an ID proponent.
He seems to have built an entire career on being wrong.
This anecdote demonstrates the difference between postulation and empirically-based claims. One may postulate that there is a remote possibility of something occurring, or they may make claims backed by evidence.
This is exactly why I recommended that you study examples of "simultaneous evolution" that would show up in the fossil record.
Rather, if evidence is brought, the evidence is confined exclusively to the possibility that evolution could provide a plausible, naturalistic explanation for the world around us.
No. This is why you had to try to think up a question for which, by its nature, any answer no matter how correct would be unsupported by evidence.
Okay, maybe my sequencing of events was not perfect, but please notice something about all of this: besides from some difficulties with the actual steps, one must realize that the steps are mere possibilities (if that), and really only serve as an escape device for evolutionists.
Alternatively, your unevidenced speculation that DNA and RNA would both be required is nothing but an "escape device" for creationists; especially as we can observe (in RNA viruses) that RNA is quite capable of performing the function of DNA.
But after all what is creationism but one big escape device: "We can't think of a mechanism ... so God did it by magic!"