In a large skillet, heat the olive oil over medium-high heat. Season the salmon fillets with salt and pepper, then place them skin-side down in the skillet. Cook for about 4-5 minutes until the skin is crispy, then flip and cook for an additional 3-4 minutes until cooked through. Remove the salmon from the skillet and set aside.
In the same skillet, add the butter and minced garlic. Sauté for about 1 minute until fragrant.
Add the orzo to the skillet and stir to coat in the garlic and butter. Pour in the chicken broth and bring to a boil. Reduce heat to a simmer, cover, and cook for about 10 minutes, stirring occasionally, until the orzo is tender and has absorbed most of the broth.
Stir in the lemon zest, lemon juice, cherry tomatoes, and spinach. Cook for an additional 2-3 minutes until the spinach is wilted and the tomatoes are warmed through.
Return the salmon to the skillet, nestling it into the orzo. Cook for another minute to heat through. Garnish with fresh parsley before serving.