Salsa vs. salsinha
Both salsa and salsinha refer to the exact same herb, parsley. The difference is primarily grammatical and regional. Salsa is the root word, while salsinha is the diminutive form. Usage varies significantly between Brazil and Portugal.
Salsa
A2This is the standard, dictionary definition of the herb. It is the primary term used in European Portuguese (Portugal). In Brazil, it is used on product labels or distinct phrases, but it is less common in spoken language than the diminutive version.
A salsa é muito usada na culinária portuguesa.
(Parsley is widely used in Portuguese cuisine.)
Comprei um maço de salsa no mercado.
(I bought a bunch of parsley at the market.)
O chá de salsa tem propriedades diuréticas.
(Parsley tea has diuretic properties.)
Salsinha
A2This is the diminutive form of salsa (literally little parsley). It is the overwhelmingly preferred term in spoken Brazilian Portuguese. It conveys a sense of the ingredient being chopped, used as a garnish, or simply reflects the Brazilian tendency to use affectionate diminutive forms.
Pique a salsinha bem fina antes de servir.
(Chop the parsley very finely before serving.)
Vou jogar um pouco de salsinha por cima da sopa.
(I am going to sprinkle a little parsley over the soup.)
Você prefere coentro ou salsinha?
(Do you prefer cilantro or parsley?)
Summary
In essence, they are the same plant. If you are in Portugal, use salsa. If you are in Brazil, salsinha is much more natural for everyday cooking contexts, though salsa is grammatically correct in both regions.