The sentence is understandable, but doesn't sound very natural.
In Chinese, it's unnatural to use two attributive-的 phrases to modify the same noun. In this case, 他们的 and 盐一样咸的 are the attributive 的 phrases, which modify the noun 咸菜. It's natural to say: 尝尝他们的咸菜 or 尝尝盐一样咸的咸菜, but simply combining the two phrases together(尝尝他们的盐一样咸的咸菜) looks clumsy(I am not sure if I should say it wrong, but at least it appears to be the bad writing.).
Another example: 展示我的衣服 and 展示美丽的衣服 are natural, but 展示我的美丽的衣服 doesn't sound quite natural.
Now, the question becomes how we can resolve this when we have to use two attributive-的 phrases to modify the same noun? We can engage 指示代词 （demonstrative pronoun）那，那个，那种，... and etc. Your original sentence can be paraphrased as: 尝尝他们那种像盐一样咸的咸菜. It's a common sense that 盐 is the most salty substance, so 咸 here can be implied： 尝尝他们那种像盐一样的咸菜.
And 展示我的美丽的衣服 can be paraphrased as 展示我那件美丽的衣服.
Sometimes, 那种 and 那件 can be omitted, so 尝尝他们盐一样的咸菜 and 展示我美丽的衣服 are acceptable. I suggest you stay with the full version when you are unsure if it's droppable from your sentence.
Hope this can help you.