In general, it is impossible to know for sure, but we can do some detective work. My two go to web sites:
https://www.space-track.org -- Catalog of all space objects kept by USAF/JSPOC
https://planet4589.org -- Jonathan McDowell's amazing catalog of all things space
AS you point out 2010-028* is the international designator for all the objects related to the launch of the Picard satellite on 15 June 2010. Space-Track has 6 listed (A-F). The international Designator is always associated with the launch itself, regardless of when the object became a standalone object (e.g. from a deployment or break up). We can conclusively say that all six objects are related, at least according to JSPOC. You will notice that objects A-E have consecutive NORAD catalog numbers (36598-36602) that means they were cataloged at around the same time. And in this case "same time" likely means the same day, since object 36603 was a Soyuz that launched that same day. So the debris objects, if they are debris, likely came off the rocket, and since there are only two, they are unlikely to be from a breakup. Object F has a catalog number of 36827 which implies it was deployed about two months later
Unfortunately the JSPOC doesn't spend a lot of time assessing what is what, and since most 3rd party sites like in-the-sky get their data from JSPOC, you end up with totally not helpful names like SL-24 DEB. In this case I believe SL-24 stands for "Space Launch 24" a Western designator for the Russian/Ukranian Dnepr rocket.
Luckily Jonathan is a little more thorough in his research and identifies the following in his version of the satellite catalog:
S036598 2010-028A Picard Picard
S036599 2010-028B PRISMA-Mango Prisma MAIN
S036600 2010-028C R-36M2 DS R-36M2 DS
S036601 2010-028D Gazodinamicheskiy Ekran GDS (Gas dynamic shield)
S036602 2010-028E Dnepr Platform A/Fairing -
S036827 2010-028F PRISMA-Tango Prisma TARGET
Here Jonathan looks to have done some additional research.
Objects A is Picard. Object B is PRISMA, which consisted of two satellites that launched attached to each other and separated on 12 August 2010 into its two independent components dubbed Mango (which kept the original designator and catalog ID) and Tango, which gained the F designator. This all matches what is in Space-Track.
Object C is the third stage of the rocket. R-36M2 is the stage name. R36 was the Soviet designator for the original ICBM that Dnepr is derived from.
That leaves us with objects D and E which are labeled as debris by Space-Track. 8 other entries in Jonathan's catalog have the word "Gazodinamicheskiy" all associated with Dnepr rockets.
Some Google searches on "Gas Dynamic Shield" led me to this Spaceflight 101 web page. It states that the shield is "element of the payload stack, protecting the satellite after separation of the payload fairing, especially during the operation of the third stage that fires its engines backward, pulling the stack into orbit"
Which means I leaned something new. I found this interesting tidbit on NASASpaceflight.com: "Originally designed to fine-tune the trajectories of multiple independently-targeted warheads, the unit has its engines mounted facing forwards." So that explains the heritage of the strange "backwards" configuration of the third stage. And the need for the mysterious debris that started your inquiry.
For this launch Dnepr used a small type 3.4.1 fairing which appears to be around 5 meters long. I would expect the shield to be slightly smaller than that, so your object is probably 4-4.5 meters. I couldn't find any pictures that conclusively would tell me the color of the gas shield.