Are /ɑːɹ/ (as in "start") and /æɹ/ (as in "parody", "marry", or "clarity") allophones? It seems that the latter can only occur when the /æɹ/ precedes a vowel in the word; i.e., when the /ɹ/ would be pronounced in a non-rhotic accent even if there are no following words. I speak a variety of American English where /æɹ/ is pronounced /eɹ/, so it isn't intuitively clear to me whether /ɑːɹ/ and /æɹ/ actually are or aren't allophones.
The other answer is correct. They are not allophones. But if you want even more proof, in the form of a minimal pair, the two words tarry /ˈtɑːri/, (to be covered with tar) and tarry /ˈtæri/ (to delay) are a minimal pair, where the difference in these vowels distinguishes the two meanings.
No, they are not allophones or in complementary distribution. There are several words with /ɑːɹ/ followed by a vowel. The word "starrer" is pronounced /stɑːɹə/ in British English (Lexico), not */stæɹə/. I'd imagine "sparrer", "starring", "scarring", "charring", "tarring", etc. also have an /ɑːɹ/ followed by a vowel. All these words are derived from by adding -er or -ing; I'm not sure if there are other examples as well that aren't so derivative from another word ending in /ɑːɹ/.