In general, there is no right under the US Constitution not to have evidence that has been voluntarily provided for another purpose not used for further criminal investigation.

The premise of the question is that the evidence was turned over:

with the reasonable, good-faith assumption that the use of that evidence will be limited in scope to the particular investigation/trial.

But such an assumption is not reasonable, and the courts will not honor it. Unless there was a specific written promise limiting the use of the evidence, or a specific policy imposing such limits at the tiem of collection, the evidence can be used in future unrelated proceedings. In general, if a person gives unforced consent to a search or an examination of evidence, that person has waived any Fourth or Fifth Amendment rights, the search or examination is lawful, as is any evidence resulting directly or indirectly from it.

Some limited exceptions:

1. Grant of Immunity When a person is put under oath and asked questions, and refuses to answer based on that person's right not to self-incriminate under the US Fifth amendment, and is then given a grant of immunity, the grant will bar the future use of that testimony against the person giving it, within the limits of the grant. The exact limits and what if any future use may be made of the testimony is defined by the terms of the grant of immunity, and may vary a bit.

2. Civil fingerprint images When a person provides fingerprints in certain non-criminal circumstances, including as part of a professional licensing process, as part of approval for th4e adoption of a child, as part of approval for the person to work with children, or as part of a background check for access to classified information, then the fingerprint data is stored in a separate "civil" database and is not subject to random or general check (that is, non-probable-cause-based checks). See this answer for more on this.

As to the specific situations mentioned in the question:

1. Rape Kit DNA There is no federal constitutional right under current case law to exclude such evidence, nor does there seem to be any state law in any US state requiring such exclusion. There might be local policies not to record such info in a database or retain it beyond the need for elimination. Something like the "civil ID database" solution could be devised, but does not seem to have been implemented as yet, although there are policy arguments for doing so.
2. Dashcam video I question the likelihood of this being reviewed for a speeding infraction. It would indeed require some significant technical analysis to accurately determine the speed at which the car was traveling, unless the video also showed the car's speedometer. However, if the video showed a more serious and clearer offense, such as the driver involved in a hit&run accident, teh video would probably be admissible unless there had been an explicit limitation of consent, acceptedm in writing by the law enforcement officer who accepted the video.
3. Parole violation I suspect that in practice the parolee would get a warning, rather than a revocation hearing, because law enforcement generally would like to encourage parolee's to report crimes, rather than to conceal them. But I am fairly sure this is a matter of individual discretion, not of mandatory policy or state statute. The relevant officials could proceed with revocation if they thought it a good idea. If the parolee's lawyer at a hearing brought out the circumstances, the hearing officer might take that into account and not revoke the parole.

In general, there is no right under the US Constitution not to have evidence that has been voluntarily provided for another purpose used for further criminal investigation.

The premise of the question is that the evidence was turned over:

with the reasonable, good-faith assumption that the use of that evidence will be limited in scope to the particular investigation/trial.

But such an assumption is not reasonable, and the courts will not honor it, because it is an incorrect understanding of the law. Unless there was a specific written promise limiting the use of the evidence, or a specific policy imposing such limits at the time of collection, the evidence can be used in future unrelated proceedings.

In general, if a person gives unforced consent to a search or an examination of evidence, that person has waived any Fourth or Fifth Amendment rights, the search or examination is lawful, as is any evidence resulting directly or indirectly from it.

Some limited exceptions:

1. Grant of Immunity When a person is put under oath and asked questions, and refuses to answer based on that person's right not to self-incriminate under the US Fifth amendment, and is then given a grant of immunity, the grant will bar the future use of that testimony against the person giving it, within the limits of the grant. The exact limits and what if any future use may be made of the testimony is defined by the terms of the grant of immunity, and may vary a bit.

2. Civil fingerprint images When a person provides fingerprints in certain non-criminal circumstances, including as part of a professional licensing process, as part of approval for th4e adoption of a child, as part of approval for the person to work with children, or as part of a background check for access to classified information, then the fingerprint data is stored in a separate "civil" database and is not subject to random or general check (that is, non-probable-cause-based checks). See this answer for more on this.

As to the specific situations mentioned in the question:

1. Rape Kit DNA There is no federal constitutional right under current case law to exclude such evidence, nor does there seem to be any current state law in any US state requiring such exclusion. There might be local policies not to record such info in a database or retain it beyond the need for elimination. Something like the "civil ID database" solution could be devised, but does not seem to have been implemented as yet, although there are policy arguments for doing so.
2. Dashcam video I question the likelihood of this being reviewed for a speeding infraction. It would indeed require some significant technical analysis to accurately determine the speed at which the car was traveling, unless the video also showed the car's speedometer. However, if the video showed a more serious and clearer offense, such as the driver involved in a hit&run accident, the video would probably be admissible unless there had been an explicit limitation of consent, accepted in writing by the law enforcement officer who accepted the video.
3. Parole violation I suspect that, in practice, the parolee would get a warning, rather than a revocation hearing, because law enforcement generally would like to encourage parolees to report crimes, rather than to conceal them. But I am fairly sure this is a matter of individual discretion, not of mandatory policy or state statute. The relevant officials could proceed with revocation if they thought it a good idea. If the parolee's lawyer at a hearing brought out the circumstances, the hearing officer might take that into account and not revoke the parole.