It would take a significant pressure drop from another appliance to make a difference on the gas logs, as they don't use much gas, its not likely that is the problem.
It is more likely that the thermocouple or thermopile is defective, does your thermocouple have two wires on it or is a capillary tube that connects to the valve ?
Have you tried to replace the thermocouple ?
What does the pilot flame look like ? It should be blue and cover most of the thermocouple.
OK thats not a great flame, you can disassemble the pilot and clean out the pilot orifice it has a very small hole in it, I usually take a bristle off of a wire brush and run it though the hole, it looks like that could be your problem, between that and maybe a new thermocouple it will probably solve your problem.
I agree that the thermocouple can be the problem, it doesn't look like its overheating from that flame though.
You should be able to get the thermocouple from a hardware store or Home Depot.
the above expert has opt out but i think you ought to try cleaning the orfice first this sounds more like the orfice is dirty or partially plugged
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is there a downdraft when you see this going on have you checked the thermal coupler yet
its anybodys guess but at least its working and you taught me something thank-you
and good luck to you