Your question: Can I also write in Spanish: Menudo caminamos en el parque. Answer: No, menudo must be preceded by an 'a' if you want to say 'often' (in fact, in Argentina we call 'menudos' what people never it from chickens, lol). There are other ways to say it. If you just wanna say 'caminamos en el parque', you can end say: 'Camiamos por el parque con frecuencia', o 'Caminamos por el parque a menudo'. Both are right. Hope it's clear. If not, feel free to tell me. REF.: I'm a Certified Spanish Teacher.

"A menudo paseamos/vamos a pasear por el parque" is more correct Pasear (v.): go for a walk, stroll