Why is ocean salinity not a good chronometer?
A good chronometer is one that satisfies the following conditions:


We know exactly how much it had of something when it began
We know exactly the rate of the accumulation of the thing, and the rate is constant.


For example, U-Pb dating in zircon is an excellent chronometer. We know exactly how much Pb is in zircon when it crystallises (= none) and we know exactly the rate of Pb accumulation by radioactive decay of U (with two separate isotopes, not one! Rejoice!).

On the other hand, sodium in the ocean is a terrible chronometer. First of all, we don't have full understanding of how and when the ocean formed, let alone how much sodium they contained in the beginning. In contrast to Pb in zircon that starts with zero and goes up, the amount of sodium in the ocean is a function of many variables. These variables change with time. How much sodium do rivers input into the ocean? Depends. During orogenic periods quite a lot. During other periods less than a lot. Volcanic eruptions? Periods such as the Siberian or Deccan traps surely input a lot, but what about other periods? Removal of sodium is also non a constant process. As bon said in the comments, you have mass removal events such as the Messinian crisis. Sodium lost in ocean floor alteration is a function of how much seafloor spreading you have at that time, and the rest of the ocean chemistry. All this changes throughout geological time. The rate is not constant!

The paper that you linked to from the Institute for Creation Research has a long list of inputs and outputs of sodium and attempts to quantify it. First of all, I would be skeptical of anything coming out of there, because religiously motivated people tend to cherry-pick the data that fits their belief. But ignore that, even if their data is 100% correct, it is still correct for today. For now. As mentioned before, the input and output fluxes change drastically over time. It is impossible to assume a constant rate for that!

Quoting from that paper:


  It is God’s world that we are studying by scientific 
  methods. It is God’s handiwork that we are learning 
  about through those studies. God deserves to be 
  praised  and  honoured—and  believed—for  what 
  we  have  been  able  to  learn  about  his  world.  If 
  our careful study of God’s world brings us false 
  or unreliable information, then what can it mean 
  for the Psalmist to sing, “The heavens are telling 
  the glory of God; the skies proclaim the work of his 
  hands.”?


Other than the fact that this passage does not belong in a scientific paper, it mentions the scientific method. Well, they fail at the scientific method because they make assumptions that make their entire study invalid.



2019 EDIT

A new open access paper shows that quite a lot of chlorine ends up in subduction zones, and removed from the oceans. As there are different subduction fluxes throughout geological time, this is another reason for salinity being a bad chronometer.