1. Freezing cold (4 °C) beverages do not increase the secretion of the gastric acid more than lukewarm or hot beverages, according to this study:
McArthur KE et al, Gastric acid secretion, gastrin release, and gastric emptying in humans as affected by liquid meal temperature (PubMed).
Coffee (360 mL) was infused into the stomach through a nasogastric
tube at 58 (steaming hot), 37, or 4 degrees C (ice cold). Intragastric
temperature, measured by an intragastric temperature sensor attached
to the nasogastric tube, returned to body temperature 16.7 +/- 2.7 min
after the hot meal and 23.8 +/- 1.1 min after the cold meal. Gastric
acid secretion increased after hot, warm, and cold coffee but the
initial temperature of the meal had no effect on gastric acid
2. Cold beverages slow down stomach emptying, that is passing of food from the stomach into the small intestine, according to this study:
Collares EF et al, 1981, Gastric emptying in children. I. Influence of the temperature of a hydration solution for oral use (PubMed)
Each baby had measured its gastric emptying for two different
temperature solutions, approximately 27 degrees C and 4 degrees C,
room and cold temperature...The results showed a significant larger
gastric retention for low temperature one.
This suggests that drinking cool water after the meals can slow down the digestion a bit and possibly cause an uncomfortable feeling of prolonged stomach fullness. This could be what the OP's parents referred to as "blockage."
As mentioned in the first study above, freezing cold (4 °C) coffee changes to the body temperature in about 20 minutes after it reaches the stomach, so it should not "freeze" the fats in the same sense as it can on the plate.
In conclusion, from the limited evidence presented above, it seems that cool water drunk with meals might cause some stomach discomfort after the meals. Someone would need to be quite observant to become aware of such feelings.