Bleach in no way adds phosphates. Now that the pool store has straight up lied to your father you can use that as a wedge to open the conversation.
When you talk about BBB the thing your father hears is "Use laundry detergent in the swimming pool." It's not his fault, it's just a misunderstanding of what the BBB thing is really about. It's about reliable testing coupled with a true understanding of what these chemicals are and how they work.
The only thing wrong with your father's pool is he has too little chlorine in it and the algae is able to persist. If you could get an accurate CYA test there would be more specific recommendations.
Below is a few excerpts from posts from Chem Geek regarding various pool chemicals and their equivalents. Perhaps some of this information could help your father see through the misinformation he is receiving. Remember - BBB is about testing and science - the fact the grocery store happens to sell the chemicals we need at cheaper prices is just a bonus.
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FROM CHEM GEEK
Bleach and Chlorinating Liquid
Let's take bleach and chlorinating liquid, for example. 6% bleach is 6% sodium hypochlorite by weight and is 4.7% sodium chloride salt. This relationship comes from the manufacturing process (called "chlor-alkali") which creates chlorine gas and sodium hydroxide via electrolysis and then adds the chlorine gas to (purer) water containing sodium hydroxide. This is done in two steps as described in order to reduce the amount of salt in the water since the water used for initial electrolysis has very high salt content.
Cl2(g) + 2NaOH ---> NaOCl + NaCl + H2O
Chlorine Gas + Sodium Hydroxide ---> Sodium Hypochlorite + Sodium Chloride + Water
There is some excess sodium hydroxide (lye / caustic soda) leftover from the above process that varies by brand with Clorox Regular having around 0.63% by weight (using a pH of 11.9). Clorox adds a small amount of sodium polyacrylate (a polymer that prevents minerals in wash water from reacting with chlorine and yellowing the clothes) in an amount that they do not disclose but is probably very small by weight. The rest, 88.7%, is water.
Other brands of bleach often have more excess lye in them with some having a pH of 12.5 implying 1% by weight of sodium hydroxide. Other than this difference (and some bleaches not having sodium polyacrylate), they are all the same. There are also weaker bleaches where most off-brand Ultra bleaches are 6% but off-brand Regular are 3% or less. Bleach and chlorinating liquid are the same except for strength and differences in the amount of excess lye.
[EDIT] You do not want to use scented bleaches or specialty bleaches such as Clorox ProResults Outdoor Bleach Cleaner or Clorox Plus Splash-less Bleach which has thickeners in it. [END-EDIT]
If you look at a bottle of Clorox Regular unscented bleach, you will notice that it says "5.7% Available Chlorine" on the bottle in addition to the ingredients of "6% Sodium Hypochlorite". This is because the product has been registered with the EPA not only as a pesticide, but has also been tested and approved for use in swimming pools (EPA registration number 5813-50 with MSDS here ). This process of testing and approval has little to do with the inert ingredients. It has to do with the efficacy of the product to kill bacteria as described in EPA DIS/TSS-12. You can read a bulletin from Clorox on the use of one of their bleaches in swimming pools here.
Baking Soda vs. Alkalinity Up
As described in this link, "ARM & HAMMER® Baking Soda is 100% pure sodium bicarbonate." It is also used in baking so is ingested in food. If anything, it is of even higher purity and quality than an Alkalinity Up product, all of which are sodium bicarbonate.
Washing Soda vs. pH Up
As described in this link, "ARM & HAMMER® Super Washing Soda is 100% pure sodium carbonate." Many pH Up products are sodium carbonate, though some are sodium sesquicarbonate which is a double salt of sodium bicarbonate and sodium carbonate, while a few are sodium bicarbonate.
Borax vs. Borate products
As described in this link, "20 Mule Team® Borax is comprised of 99.5% pure borax, a naturally occurring mineral composed of sodium, boron, oxygen, and water. (The scientific name for borax is sodium tetraborate decahydrate.) The remaining 0.5% is composed of trace minerals."
Products such as ProTeam Supreme contain Sodium Tetraborate Pentahydrate which is the same as Borax except it contains less water (hydrates are chemicals that contain water including those in solid form in a crystal). ProTeam Supreme Plus and ProTeam Gentle Spa do not disclose their "Proprietary ingredient", but it is most surely Boric Acid given that the product overall has a pH of 6-8, that it increases borates, and that the product also contains some sodium tetraborate presumably to make the pH more neutral (boric acid is slightly acidic).
Muriatic Acid
This is diluted Hydrochloric Acid where full-strength is usually 31.45% (see this MSDS) and half-strength around 15%-16%. There is no difference between brands and the inert ingredient is water.
Calcium Chloride vs. Calcium Hardness Increaser
Dowflake is around 96% Calcium Chloride Dihydrate. This MSDS gives 77-80% Calcium Chloride, < 5% Sodium Chloride, 2-3% Potassium Chloride, 15-20% Water. This MSDS gives some slightly different composition for this product (a little more calcium chloride content with less water).
Peladow as described here is at least 91% Calcium Chloride Anhydrous with 4.5% Water, 2.5% Potassium Chloride, 1.5% Sodium Chloride. This MSDS gives some slightly different composition for this product.
Hopefully this gives you some idea of how there really is no significant difference between grocery or big-box store vs. pool store chemicals, at least for the common ones used above and how their inert ingredients are truly not a problem.
FROM CHEM GEEK :
For every 10 ppm Free Chlorine (FC) added by ANY source or chlorine, it will also add 8.2 ppm salt when the chlorine gets used up since it gets converted to chloride (which is salt). In addition, bleach, chlorinating liquid, lithium hypochlorite and chlorine gas all add an additional 8.2 ppm salt upon addition whereas Trichlor and Dichlor do not (Cal-Hypo is in between adding 2-4 ppm extra salt). Salt is far less of an issue than the Cyanuric Acid (CYA) added by Trichlor and Dichlor or the Calcium Hardness (CH) added by Cal-Hypo and on a percentage basis it rises the least. The concentration of product is not relevant towards these numbers that are relative to the FC.
The following chart shows how much the CYA, CH and Salt would rise after 6 months if there were 2 ppm FC per day chlorine usage and no water dilution:
........................ CYA ... CH ... Salt
Trichlor .......... 220 ...... 0 .... 295
Dichlor ........... 325 ...... 0 .... 295
Cal-Hypo .......... 0 .... 255 ... 360
Bleach* ............. 0 ........ 0 ... 595
*Bleach, chlorinating liquid, lithium hypochlorite and chlorine gas all result in no CYA, no CH and the same amount of salt.
Note that a rise in CYA of 220 ppm is much worse than a rise in CH of 255 ppm which is worse than a rise in salt of 595 ppm.
