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We study the Laplacian flow of a $\mathrm{G}_2$-structure where this latterstructure is claimed to be Locally Conformal Parallel. The first examples oflong time solutions of this flow with the Locally Conformal Parallel conditionare given. All of the solutions are ancient and Laplacian soliton of shrinkingtype. These examples are one-parameter families of Locally Conformal Parallel$\mathrm{G}_2$-structures on rank-one solvable extensions of six-dimensionalnilpotent Lie groups. The found solutions are used to construct long timesolutions to the Laplacian coflow starting from a Locally Conformal Parallelstructure. We also study the behavior of the curvature of the solutionsobtaining that for one of the examples the induced metric is Einstein along allthe flow (resp. coflow).

Solutions of the Laplacian flow and coflow of a Locally Conformal Parallel $\mathrm{G}_2$-structure